At the time the English (they will not be British until 1707) and the Dutch are embroiled in the War of Spanish succession. Besides, George I will not become King of Britain until 1714 - Queen Anne did not have his ambitions on Swedish Bremen that her successor had. She did not have the associated hostility against Sweden and did not send the Royal Navy into the Baltic to try to provoke an incident that could cause Britain to enter the war as Hannover had.
Archangelsk is not on the Baltic Sea so this is irrelevant. OTOH, Archangelsk was the only Russian port and all foreign trade (English & Dutch) was going through it. Swedish possession of that port would cause noticeable problems to the traders.
Well, it is not like, especially with all that program of yours, Swedes would be able to dedicate significant resources to capturing and holding Archangelsk. OTOH, if they are lucky, this makes peace with Russia close to impossible unless its return is a part of the peace conditions. So, basically, you are making situation worse, not better for Sweden.
I will admit that my knowledge of internal Russian politics at the time is lacking, but I understood that some of the conservative opposition, however neutered, gathered around Peter's son Alexej, but since he was lazy (or at least not energetic enough for the tastes of his father) and rather initiativeless, nothing came of it.
Well, as you said, your "knowledge of internal Russian politics at the time is lacking".
This is a rather popular BS generated by Peter's propaganda (to justify Alexei's murder) and then repeated for the following couple centuries because it provides a clear cut black and white picture (Peter I - reforms, Alexei - growing beards; Peter III - drunkard, Catherine II - reforms). There was no "conservative opposition" worth mentioning and there is no real indication of Alexei's excessive attachment to the "old ways" (which he could not know). He traveled extensively, had a better education than his father, actively participated in his wars while still a teenager, chose himself a foreign bride, etc. He most probably did not approve everything his father was doing but his father was doing things in an extremely wasteful way and if someone is not very fond of an excessive drinking, this does not qualify a person as a reactionary. Alexei was simply set up by his step-mother and her supporters and this has very little to do with his views.
As for his alleged laziness, it is rather rather hard to say now if this was the case but comparing to his father who had a never-ending itch in his posteriors any person who was thinking before acting could pass for a lazy one.
You are right that Sweden lacks the power to take and force Russia to yield, but if things become bad enough for Peter, he might seek peace to return later.
Well, he did try more than once but Charles was not interested.
By 1704 the Baltic provinces had not been as ravaged by repeated Russian raids as they had been 1708, which was why Karl XII elected to invade Russia rather than go north - he considered Estonia and Livonia unable to sustain his army.
And yet, Levenhaupt was sent there to raise an army and supply train. Anyway, he was marching from Saxony and going to Latvia did not make too much sense if he was trying to advance deep into the Russian territory. Anyway, soon enough he encountered the same supply problems.
In 1704, the situation was not that bad yet, and moving north to retake Nöteborg and Nyen and destroy the embryo of Saint Petersburg was a viable option, especially if Czar Peter could be compelled to fight for his new capital and be soundly defeated there.
This would be quite possible with an adequate force but there was very little for Peter to fight for at St-Petersubrg site: city was on the initial stages of its development and it took decades for it to grow into a great city. Few wooden houses could be rebuilt at any time. So no, unless there is a close to 90% chance of success, Peter would not be compelled to fight a serious battle over its possession. Nyenskans (see below) was a small fort destroyed by 1704. Nien was a city it had been protecting so there was not too much to take there. Noteborg (taken in 1702) also was a small (even if important) old fortress (2nd picture). It was important because it had been controlling entry to Neva from Ladoga Lake. In other words, both places made sense only as a way of not letting Russians to get to the Baltic coast.
OTOH, fortifications of Pskov and especially Novgorod had been upgraded in 1701 and Charles XII is not known for his ability to take the fortified places.
The Russian army was still developing in 1704 (it was steadily getting better, but was not the force it would be 1710 onwards in 1704).
If Peter is captured or killed, a lot of his accomplishments could be lost in the ensuing chaos - if he survives, he is shrewd enough to restore his power, even against the heaviest opposition, but might consider Sweden too tough an opponent for Russia to take on alone, at least before more of his reforms have borne fruit. The Crimeans or Ottomans might be his next target instead.
Well, let's keep to a practical side of a history because conveniently killing that or this personage can change quite a few things.
As I said, Peter would not risk a major battle without an overwhelming chance for success and, if risk is too high, he would be probably absent from the theater because (even if I intensively dislike him), unlike Charles, he was not considering his main task riding at the head of his troops.
Once, after getting the glory of Poltava, etc., he became dangerously adventurous and it ended up at Prout (where he panicked in a situation when any half-decent Russian general starting from 1740 would march ahead and defeat the Ottomans) so feel free to offer scenario under which he surrendered to the Ottomans.
So the chances of what you wrote are extremely low.
So the war would continue until both sides are too exhausted to continue. Taking into an account personalities of both leaders, this would be even longer affair than in OTL ending, in the best case scenario, with the minimal Russian possessions on the Baltic coast. I'd say, Ingria. But even for this you really need to kill Charles.