Regarding the overrall thread topic, for the record I don't think the federation is all that likely to run into significant opposition from the imperium. Yes, the imperium won't like them, they're clearly all heretics and blah blah, but the imperium doesn't like a lot of people and unlike, say, the Tau, the Federation isn't trying to manifest destiny a bunch of imperial planets. There will be conflict here and there when a segmentum commander tries to conquer some little chunk of the federation,and sometimes win and sometimes not, but it's going to be, in 40k terms, little more than skirmish and small wars (in ST terms I'd put it somewhere between the Klingon/Federation war in mid DS9 and the Dominion War).
Ditto orks, Tau, Eldar, etc. The federation is going to see its fair share of fighting but they're not going to catapult to the tip of anyone's "kill all these guys right now" lists. Most likely result is either they collapse from the constant pressure after a few centuries or they shrink down to a more defensible size as they tech up and reorganize.
In terms of Strategic mobility, raw sheer speed is only a minor facet. We're talking more about the issue of the Imperium knowing where it's ships are, relaying orders to them, them gathering and preparing for the coming battle relying on information that could be months, years or even decades old in addition to the battle group potentially arriving at different times and different locations.
You are overstating the difficulty on the imperium end, and drastically so. The imperium has FTL communications, and they're functionally enough to run a military. Imperial warp drive is not precise, but it reliably gets ships where they need to be at a roughly predictable time.
No, you can't time an offensive down to the minute or call on reinforcements to help you out mid-battle, but you're talking like the imperium will send 10 ships to take a planet, and half of them will arrive there sometime between 2 months ago and 20 years from now, with the other half ending up in the wrong system by accident, and that's not the case either.
GW invokes a lot of age of sail tropes when it comes to the warp, and that should be your guideline. So you'll know roughly how long a trip will take and roughly where they'll end up,, and some routes will be faster than others because of currents in the warp (even if the distance in real space is the same), but it's not a total crapshoot as to what happens when you give an order.
Further its unlikely that it would take the Federation 2 months to simply get reinforcements. The DS9 episode "Way of the Warrior" establishes it takes 8 weeks to reach Cestus Three which was on the opposite side of the Federation compared to the titular station. So response times for closer fleets should be days or weeks rather than months.
Depends on where they drop into real-space, really. If they drop in orbit around the planet, yes most definitely. If they emerge days' travel out from the world in question then things tilt right back to the Federation.
A response time of days still favors the imperium significantly, because imperial ships outmatch federation ones by a massive degree. You don't need to get a few ships to respond to an imperial incursion, you need a ton of ships. And the federation doesn't have dozens and dozens of ships within a two day range of every single planet, it will take time to gather them up and assemble a counterforce (also, needing only a few dozen ships to match a small imperial fleet is being very generous to the federation).
Add it gets much, much harder if you gave to respond to more than one crisis at once.
Hmm I had no idea the Romulan Empire was part of the UFP back in the 23rd Century. Oh wait it wasn't.
This is the same episode that stated the Romulans fought the Earth Romulan war with only impulse drive, and only figured out FTL drive at some point post balance of terror. You can't take every bit of the series as gospel, and particularly not TOS given the writers were still figuring out how everything was supposed to work.