Well, they definitely conflicted on and off with the Ottomans through their existence, and were the other major power of the Middle East. They weren't quite up to Sassanid standards, admittedly, but they did a pretty good job of fighting the Ottomans, so they certainly count as "a power with enough strength to be comparable regionally to Russia or the Ottomans" (bearing in mind that the Russians weren't in the region until after the Safavids collapsed).
The trouble is that Russia didn't really expand into traditional Iranian territory until the 18th century, and in any case were not devoting any juggernaut-level amount of forces to expanding eastwards. Russian efforts didn't really spread into the desert and steppe regions that were of interest to the Iranians, so what you would most likely see would be a few border skirmishes followed by a treaty, much like when Russia started to expand into China. That might change later, but certainly in the short term Russia would probably find a buffed-up Persia more useful as an ally against the Ottomans than as an enemy.
That might change if the Persian state collapses as per OTL, or as Russia and the rest of Europe overtakes it technologically, but in the short-run I would expect an alliance and reasonably friendly relations, not a great rivalry.
Well, it depends on whether the Ottomans can conquer Mesopotamia or not. If they can, then their Middle Eastern expansion is relatively unhindered; they might face a bit more trouble in exerting influence over the Gulf Arab states and Oman, but those were hardly critical or huge resource sinks to my knowledge. If they can't, then they would lose all influence over those states, which would likely remain in the Persian sphere, but they probably wouldn't have a lot of trouble conquering Egypt, Syria, and the Hejaz. Their expansion into Europe is probably relatively unaffected, because either way they have a lengthy border with Persia that they have to defend, as they did IOTL, but perhaps more so. It might end up expanding a bit less into Europe and having its gains reversed a bit faster, especially if Persia does manage to ally one of its European enemies like Russia.