Thanks for bringing up some interesting issues. Not to get into a bickering, here is continuation of a framework broken into 2 main parts: 1st, what we do know and 2nd, what is open to the speculation.
What we do know:
Alexei - personality:
1. Reasonably well-educated, unlike his father. Regarding his education, Menshikov was not his
tutor but a person responsible for general overseeing his education. AFAIK, his personal contribution amounted to enforcing the drinking habits, forcing Alexei when still a child to play a toy soldier to a delight of his father and quarreling with at least one of Alexei's tutors over the issues of an etiquette.
2. There is no
factual reason to assume that he was overly attached to the old ways: when in 1717 he fled from his father, his mistress, Efrosinia, traveled with him dressed as a man (a complete taboo from the old ways perspective), they attended theaters (further down the road to Hell) and she was attending a Catholic church to "listen to music" (Catholic church music - you are already in Hell!). How "deep" was his religion is hard to say because the same question applies to quite a few Russian monarchs before and after him. There was "a mental split" on a number of levels: (a) ruler's personal attachment to the Orthodoxy doctrine and, mostly the rituals, (b) ruler's opinion about extent of his/her prerogatives as being anointed by God and (c) view on a Russian Church as a tool in ruler's hands.
As a side note: Peter's famous "Всепьянейший Собор" was a mockery of a
Catholicism and says nothing about his Orthodoxy.
3. He was seemingly a heavy drinker, which is not a big surprise taking into an account the environment in which he was brought up: he was forced to participate in his father's drinking parties when still a child.
4. His relations with Charlotte were fluctuating from affection to a complete neglect but this was not an uncommon thing for the arranged royal marriages: look at the practices of Louis XIV or August II or even of his own father who openly had a mistress, Anne Mons, for years and then openly lived with a former POW "washer woman" (in her own definition) whom he got from Menshikov. At least to some degree neglect was due to the Peter's orders to travel somewhere.
As a side note: Marriage by the brideshow was a tradition but not a must. Ivan III married Sophia Paleolog without it. Not sure if Vasili III married Elena Glinskaya (mother of Ivan IV) following that procedure 2nd marriage of Ivan IV to Maria Temrukovna, daughter of the Princde of Kabarda happened without it. Irina Godunova became wife of Fedor without the brideshow. While Tsar Alexei passed through the formal steps, he picked up Natalia Naryshkina prior to it. So it is probably safe to say that nothing would prevent a Tsar of marrying whoever he wanted with or without following that tradition.
5. While still in Russia was assigned various military/administrative functions. Peter was not quite happy with what he managed to accomplish in the terms of fortification of Moscow but it is anybody's guess up to which degree this was a byproduct of a persistent campaign run by Catherine, Menshikov (and whoever else).
6. Seemingly not too energetic, at least in the same way as his father who simply could not be at rest (some of the modern historians consider this a symptom of a specific mental disorder preventing concentration on a subject and promoting physical activities; not being a shrink, can't say how true is this theory).
Charles XII:
1. The fact that Charles was somewhat obsessed with the notion of justice (as he understood it) and unwillingness to accept the losses is emphasized by a number of authors (the last book in which I read that was his biography "Chalres XII, or 5 bullets to the King" by B. N. Grigoriev, quite sympathetic to the hero) and it is confirmed by the diplomatic exchanges prior to Poltava.
2. Timing of Charles return to Sweden can not (IMO) change too much by the reasons already mentioned: he needed emperor's permission for crossing the HRE and until WoSS was not over, he was considered a hostile power. Actually, the Ottomans, or rather Siraskir and Khan, who became fed up with his presence, tried to negotiate his earlier return: he should cross the PLC territory (with August's agreement) under the Ottoman escort and join Steinbeck's army in Germany but by whatever reason Charles refused to move and the whole affair ended with a shootout in Bender.
3. An idea that immediately upon his return Charles could attack Russia was by that time impractical (this plan was offered by Frederick of Hesse but rejected by Charles): defense of Stralsund and Rugen (to provide communications with Stralsund) was necessary to prevent an attack on Swedish mainland. By squeezing Prussians from the Swedish territories around Stralsund and requesting return of Stettin he started an open war with Prussia, Denmark already had been hostile and Hanover started siege of Wismar. August also wanted some piece of the Swedish Pomerania. Before the siege started Charles managed to proved an adequate supply of Stralsund with provision but against 50,000 of the Prussian, Danish and Saxon troops he had in Pomerania only 14,000 including 9,000 of Stralsund's garrison. So the whole thing resulted in a protracted duel between the Swedish and Danish fleets and eventually the Danes won (it seems, not as much by thinking more Swedish ships but because Swedes had serious problems with repairs and supplies). Which provided the allies with opportunity to land on Rugen and Charles had to face 12,000 opponents (led by Anhalt von Dessau, a very prominent figure in Prussian military history) in a fortified camp with 750 infantrymen and 2,200 cavalry. His 2 night attacks on enemy's camp failed with a loss of 400 killed and 200 wounded. Rugen was lost and fall of Stralsund became a matter of time. So the whole grand schema (attributed to Charles) of a two pronged attack on Russia impossible. Actually, when capitulation of Stralsund became inevitable Charles (1st time in his career) decided to compromise: he offered Prussians Stettin and Danes a monetary compensation for not taking Stralsund and Rugen. But it was too little too late.
Speculations:
1. Military situation:
In that scenario it is somewhat more favorable to Sweden.
In OTL:
In 1712 Russian army of 48,000 under command of Menshikov entered Pomerania
In 1713 Russian army supported by Saxon artillery took Stettin and transferred it to Prussia
In 1713 Russian army defeated Stenbock and, together with the Danes forced his capitulation at Tenning in 1714
In 1713 Russian army under command of Prince M. Golitisin entered Finland, took Abo (then capital of Finland) and defeated Swedes in two battles (in 1713, 1714).
We may assume that if Alexei opted for a less aggressive approach to a war, at least some of these events would not happen. OTOH, the opposite also can be assumed: absence of a peace more or less forces Russia to continue acting aggressively thus depriving Swedes from their resources (and preventing attack on the Baltic provinces) and involving new combatants who can engage Sweden on their own (which actually took place: the Danes explicitly insisted on Russian non-participation in a siege of Stralsund). Probably an attack on Finland was more or less to be expected as well. However, if Menshikov is immediately removed then
probably Sheremetev is in charge of the Russian troops in Pomerania and Sheremetev is a much less energetic commander, which
may save Stenbock. Still, Swedes keep being seriously outnumbered and by the time of Charles return are in a bad strategic position.
So the 1st big speculative area is Alexei's grand strategy and appointments. For example, putting M. Golitisin in charge of the army in Pomerania and leaving Sheremetev at home as a "military consultant" (based upon what I read about Sheremetev, he may like such an option; at least in OTL he complained bitterly about his appointment to Pomerania and Mecklenburg in 1715 - 17).
2. Peace:
There is no visible reason for Alexei to give away the newly-acquired Baltic provinces without a major defeat and a major defeat is extremely unlikely taking into an account general situation described above: Charles returns only in 1714, upon his return no miracle happens and almost everybody in the region is fighting against Sweden so invasion of Russia is unrealistic.
Option of declaring war as "unjust" (thanks for the idea

) is not going to work because it puts to doubt a "legacy" going back to Ivan IV (1558–82), continued by Boris Godunov
(1590–95), Michael (1610-1617), Alexei I
(1656–58) and Peter.
An option "we'll return the Baltic provinces and you will not mess in Poland" also not working: giving away territories won by a heavy fighting in exchange to the promise which can be easily broken and which, anyway, would not mean too much for Russia (Russian troops are already in the PLC and Sweden is not in a good position of invading it) does not look as an equal exchange either.
So the peace has to wait until OTL Charles' death or until he is ready to accept the losses.
3. Domestic policies:
1. Rollback all the way to Domostroy, long beards and long sleeves is extremely unlikely both due to a general unwillingness of the nobility and because it does not look like Alexei had such inclinations on a personal level (see above about him and his mistress traveling in Europe).
2. It is quite possible that Alexei ends up with appointing a Patriarch instead of creating a Synod. However, experience of his (quite devoted) grandfather demonstrated that Patriarch is not a sacred cow and if he has too high opinion about his place in Russia, he can be replaced without too much hassle. Anyway, the top level of the Russian priesthood already trained to do what they are told by a ruler so a trouble is not to be expected.
3. Moving capital to Moscow is not an issue in 1710 because it is still a capital.
4. What to do about St-Petersburg? Russia got 2 major ports, Riga and Revel (plus some smaller ones including Narva). However, location of St-Petersburg has some advantages in the terms of being better adjusted to the traditional Russian trade routes to the Baltic coast so the site can be further developed as an additional trade port.
5. Fleet. Alexei told Austrians pretty much what they wanted to hear about abandoning fleet completely but most probably he would be persuaded to maintain some for the case of a future war with Sweden. While by location St-Petersburg gives some advantages as a main naval base (being protected by Kronstadt, see the map), there were distinct disadvantages of keeping the wooden ships in a low-salt water. In OTL Reval had been used as a second base but its harbor seems more open and vulnerable.
6. Wild guesses:
1. Alexei's position on the existing administrative institutions
2. Alexei's position on his father's experiments with the Russian industry/economy (mostly disastrous)
3. Alexei's position on the existing system requiring nobility to serve in the army (would he do what Peter III did?)
4. Alexei's position on expansionism (war with the Ottomans, war with Persia, "imperial" plans for Baltic domination, etc.)
5. Alexei's position on pretty much everything else.
