To start with, having progressive ideas and being able to 8mplement them is not the same, especially when most of “society” does not support these ideas. This would apply both to his expressed idea regarding equality of the religions (even a hint to it greatly simplified Catherine’s coup) and an assumption that he could do something drastic about the serfdom (Catherine tested the water and found that not just the nobles but even the merchant class did not support this idea, actually, the merchants wanted expansion of the right to have serfs to themselves).
Which does not exclude a possibility of something “progressive” being pushed through but what and how is anybody’s guess. Perhaps he could stop oppression of the Old Believers and allow the Jews to settle freely in Russia (both actions would cause complaints from the merchant class).
Peter’s main problem (as far as it is possible to tell) was a rather superficial attitude: he was assured in his divine right and did not bother to learn about the country he is ruling, neither did he show any serious interest to the finances, economy, etc. And he was clearly incapable of taking decisions in the tough situations.
Regarding Holstein, the British fears about the Russian base would make sense if (a) Russian Baltic fleet was a meaningful fighting force (which it was not at that time) and (b) Russia was a hostile power. Of course, an assumption that the British government was always acting based upon a common sense and analysis of the situation is not quite realistic. Anyway, Peter became an ally of Prussia and, IIRC, was not even at war with Britain during the 7YW, plus we had been assured more than once that under no circumstances Denmark would agree to cede a square inch of its territory

-)) so the issue is more or less mute. Probably Fritz would agree to act as a mediator and some mutually acceptable arrangement could be reached.
As far as Balta incident is involved, in OTL it was a byproduct of a convoluted Catherine’s policy, which simply may not exist under Peter. To promote “Russian party” in the PLC Catherine and Fritz agreed to chose the least powerful representative of that party who was poor and dependent upo; the Russian money. Small wonder that King Stanislav proved to be quite impotent because even Czartoryski family turned against him. Then there was an issue of the “dissidents” (rights of the Russian Orthodox) which Catherine had been trying to resolve but to which Peter would probably pay no attention. So the Ottoman War could happen at different time and perhaps with a better preparedness. The main problem was not, actually, a war but a peace: in OTL under Prussian and Austrian pressure Russia returned quite a few conquests to the Ottomans and had to reconqur them in the next war so there is a lot of potential for not making these concessions. One of the options would be a more aggressive conduct of the war on its last stage: instead of holding the line Russian army could keep advancing making the Ottomans much less comfortable.
As for the succession law, there was none except for the right to pass crown to whoever the ruler wants. But it is reasonable to assume that under the normal circumstances Paul, especially if he is a grown up person, would succeed his father.