What are the prospects for the Russians expanding on their foothold on the southern shore of the Caspian, to eventually assimilate Persia?
The key here I think would be if the Tsar could somehow position himself to be accepted by the Persians as the protector of Shi'ite Islam against Ottoman power. The barrier is that the Persians had their own dynasties that did that, thank you very much. Initially the Russians would surely be seen as infidel interlopers. It could be though that the Russians themselves, in breaking the power of the Shi'ite Persian Shahs, discredit the domestic dynasty and then if they are astute enough (a big if when we are dealing with the Romanovs!

) take up the mantle, reassure the Persians they have no designs on conversion to Christianity, and build up their reputation as protectors of Shi'ites from Sunni domination. This might give them the opening to dominate Persia, and from there move into Mesopotamia, which also has large majority-Shi'ite regions. So does the eastern, Persian Gulf shore region of Arabia.
I'm thinking this all happens after the reign of Peter the Great but before the end of the 18th century and the French Revolution, which probably would not be much butterflied by all this. The British would be aware the Russians are moving toward the Indian Ocean and be alarmed by this, but they'd have other concerns too that would distract them and make them balance this concern against others; if during the Napoleonic wars for instance Russia and France get at odds as they did OTL, the British are going to welcome Russian alliance and set aside many quarrels they have with them.
Quite obviously, oil has nothing to do with any of this; the Russian goals are to opportunistically expand into an important territory that borders their current conquests and then gain a warm-water port for naval ambitions. They can pretty well get control of the whole Persian Gulf this way; their next hurdle is to secure passage out of its straits, which means either coming to an understanding with Oman or conquering it, both of which the British (when they are not allied with Russia) will be intriguing to prevent. But even if the British keep control of Oman they don't control the other shore, they can't stop Russian ships from using the strait without being at open war with them.
Now if the Russians can do this and the dynasty survives and keeps control of the Shi'a lands into the 20th century, then all of a sudden the oil becomes a big issue. At that point, it all depends on how firmly the various Shi'ite people under Russian control these past couple centuries identify with the Russian regime; if their allegiance is weak then intrigues of other European nations to split them off will be rife and could lead to big continental or world wars, or certainly become a front of them. If the Shi'ite lands are loyal to the Tsar, or some successor Russian regime, then mission partially accomplished.
As for getting onto the Med--well, a long-standing Russian goal is to get control of the Dardanelles, particularly Constantinople. They've been frustrated first by Ottoman power and later by other Europeans, notably the British, seeking to deny them this access. It doesn't seem beyond the realm of possibility that there are plausible timelines where they succeed, and perhaps ally with Greece to get a firm foothold on a small part of the Med that way, again gaining them the benefit of their warm water ports on the Black Sea, making them in effect ports on the Med--in fact Constantinople itself would be the main port, if they don't also get Greek ports like Salonika.
In OTL of course they accomplished neither of these things, but the question is, what stopped them, and how hard would it have been for things to go otherwise. In particular if they could reach either the Persian Gulf or the Aegean well before 1800, I daresay that even in the 18th century the British would have been ill situated to stop them, and Russian presence on these shores would be a fact they'd have to live with.
I think placing themselves as defenders of the Shi'ites would be unlikely and difficult, but if it were done, Russian power would be leveraged by the aid of the Shi'ite peoples themselves, so the question of overextension which otherwise would be a very pointed one is to some extent addressed. By becoming the champions of a minority sect of Islam they'd probably intensify their problems with the other Muslim peoples they face, in Central Asia particularly. They'd have to defend their Persian holdings against Afghan raiders and the whole steppe flank would be that much more hostile against them.
Holding to the idea that the Russians have limited resources and have to pay for this sort of success with falling short elsewhere, I suppose we can assume they are somewhat less successful at securing Central Asian territory and eastern Siberia; their foothold on the Pacific is thinner. They'd still control the northern tier of Siberia, probably by default, unless they fail to secure vital routes that run southward.
Also, these moves of theirs would tend to assume they had some naval ambitions, and having invested and sacrificed to gain access to two warm-water seas, they probably would follow through and develop navies and merchant fleets operating out of these ports. This again sets them at odds with the British. It is by no means a foregone conclusion that the British will pen them up in their ports, but to break out of British encirclement Russia would have to invest a lot in its naval enterprises and it is not clear just why they would or how it would pay off. Perhaps they can set themselves up as a rival merchant power, but the British do have a long head start. One reason British merchant ships reigned supreme on the high seas was that the RN had established itself as the supreme naval power; with two navies of comparable strength at odds with each other the seas would simply be less safe for either side, transport would be more costly, and global development would be slowed down considerably.
It seems fairly likely then that if they did manage these two gains, control of the Dardanelles and access to the Persian Gulf by say 1800, that they would still wind up pretty much conceding mastery of the high seas to Britain. Control of the Dardanelles would then basically mean security on the Black Sea, which would be a Russian lake albeit with some hostile shores, while the Persian Gulf would be a trading port giving the Russians some cut in a portion of world trade even if its mainly British ships doing the carrying. If the Persian Gulf region is securely loyal to the regime then when oil becomes important, oil revenues will go a long way toward reviving the fortunes of Russia, which would be sitting on top of essentially all of it in the whole Persian Gulf region--Iran, Iraq, and the OTL Saudi fields as well. Plus of course Russia's OTL fields on the Caspian which OTL were important long before the Persian Gulf fields were discovered and developed!