Sadly, my knowledge of Peter I is very small,
It's a fascinating topic. There's a fairly recent biography which is very informative about his times and his Russia, if I could just find the bloody thing...
although it seems to me that if there was a Civil War in that era, both Sweden and the Ottomans would love to carve out a chunk of Russia. Not being an expert on the timer period, I will defer that point.
Oh, I absolutely agree that both will smell opportunity; but working from my 1711 scenario, I'm not certain that there's a very conveniant chunk to be carved anywhere.
This is just before the Russians go on their winning streak against the Ottomans, and their last endeavour, Peter's first campaigns in the previous century, were foiled by some pretty serious teething problems: waste, inefficiency, administrative chaos, command bickering, and poor relations with the Cossacks. In the end, the Russians did take Azov; but then, the Cossacks had done that by themselves decades before, and only withdrawn because the Kremlin was too smart to support them against Ottoman power. Russia got to keep the port in 1700 because the Ottomans had bigger problems elsewhere.
In short, as of 1711, the Ottomans were still the dominant power of the whole Black Sea shore; they'd pretty much given up on the Ukraine (and after Poltava, the pro-Russian Cossacks were in the ascendant), where their various adventures had reached the limits of where they could sustain power close to the Polish and Russian heartlands; and in OTL they batted aside Peter's attempt to attack their position without overly much effort.
In OTL, having practically reduced the tsar and his army to their hostage, the Ottomans let him go, and asked only for Azov back, which the northern-looking Peter wasted no time in giving. The reasons why did this, it seems to me, all still apply if he is an actual prisoner, or if he is a corpse and they're negotiating with his gang: they have nothing in particular to fear from Russia (yet), and also nothing in particular to gain; they're much more concerned with the Hapsburgs, with a formidable army that is starting to disengage from the French and continue its interrupted march against Ottoman overestension in the Balkans; and if there's one thing Sultans didn't like doing in the climate of Constantinople at the time, it was giving the prestige of victory - and a handy corps of loyal veterans - to the Grand Vizier, or anyone else liable to get too big for his boots.
But I do think the Ottomans will become involved in this war to some extent. How much, and for whom, really depends on hos it goes, since they have no interests at stake except keeping the Russians distracted with one-another. They might try and re-open the struggle for Ukraine, but the result of the last round, the Ruyina (which was half of the country depopulated by the march of Ottoman and opposing armies and the Poles free to move back in and party like it was 1648) had discredited them with Orthodox opinion pretty thoroughly.
Ukraine is an interesting question. The Poltava campaign has of course been misrepresented by commentators since Lord Byron

p), but it was actually little more than Hetman Mazepa taking his personal retinue and joining the Swedes in the hope that they won: the Orthodox church remained staunchly Russophile, and at least half of the Cossacks followed its lead, with the rest playing wait-and-see. (Note also that the defeat of Mazepa did very little to damage Ukraine's autonomy, which declined naturally as the other powers able to play for influence there were beaten back and the Russian state aparatus grew: the Russophile hetman Skoropadskiy was a clever man, and made himself nearly as independant as Mazepa had been.)
Speaking of church support, this is the central problem with overly enthusiastic foreign intervention: the Russian identity upheld by Alexei was deeply Orthodox and anti-foreign. Anybody who rides into the Kremlin on Protestant or Muslim bayonets will only stay there by grace of same. So that's why I prefer to have Peter killed off at a moment when his regime was still vulnerable at home, but the Swedes had largely lost the ability to go on the attack (they were out of Livonia by this time, their forces in Finalnd were a neglected mess with no cavalry and not enough supplies, and the Russians were already building the flotilla that would give them control of Ladoga).
In my opinion, since the Swedes are very unlikely to actually succeed in getting to Moscow and holding up a government (their situation is not better than that of the Poles in 1612, it would seem to me), the situation of the boyars is actually
improved as they don't have to explain why, if they hate foreigners and heretics so much, they're allied with them against the rightful tsar of Holy Russia.
The Swedes will do
something. If they're lucky, they may even get Ingria back: the Russian claim was old enough to be kosher for the boyars, but Alexei said several times that he would burn Petersburg to teh ground if he rose to power, and when it comes to sea-access beggars can't be choosers. But by this point they lack the capacity to go fooling around in Russia on a large scale. Most of their land forces were already fighting the Danes and assorted Germans, anyway, so Russia going out of the action is more of a respite than an opportunity.
(Phewf! It would appear I've gotten a little carried away; excuse me, but I do love the GNW and Peter the Great generally, which I think have vast unexploited AH potential.

)
Assuming no intervention, who will win that War? Peter I or Alexei (it has to be him, because an actual war between father and son is too good to pass up!).
Oh, it's very dramatically appropriate, I agree, and the two really did have it in for each other. They got really venomous after 1711, at which point Alexei still technically worked for dad (the work being "organising supplies for the troops in Poland" and other things that kept him as far as possible from Russia and the boyars); but with an opportunity like this, and the boyars taking matters into their own hands, I'm pretty sure Alexei would be ready to make an endeavour for the throne in 1711. Let me find that damn biography...
I'm really not sure who'd be the winner, to be honest. Russia was such a bubbling cauldron at the time that it's hard to calculate much about it. I think you could justify things either way, so do what you like!
And on Alexander II, I'm surprised everybody is saying that it would not be an even civil war without foreign intervention. Which faction would the military support, the Czar or the nobles?
That's another interesting question. The Russian army, before the reforms of Milyutin in the 1870s, was... a
strange institution. My prediction would be that various officers would lean their own ways depending firstly on the size and solvency of their estates, and take their men with them. It would be a mess.