Alright can we stop arguing about what kind of political entity it is and focus on the threqd?
Sure. So getting to the polities in general, my thoughts are:
As I posited, I think the Federation will do very well against chaos as they simply have the right mindset to fight it and are accustomed to dealing with extradimensional invaders and shutting them down. I do not see chaos cults making much headway given the Federation lacks most of the things the Imperium has that promote chaos, and the Federation is going to be both very interested and very good at analyzing the myriad things that shut down the Warp and seeing if they can't replicate it across their space.
I see Orks as the biggest threat. The Federation
is going to try to negotiate with them in good faith before realizing Orks are genetically incapable of peace and that's going to bite them in the ass hard. It's possible the Federation can indeed whip up some technobabble solution to the spores but I find it somewhat unlikely and once infested, it's really hard to remove Orks, and the Federation is really just
crap at dealing with strong in-your-face brutal infantry attacks.
The Eldar and Dark Eldar are going to go as the Orks, but not as effectively because they're not going to be able to infest a world forever. In both cases I think the Federation will try to take precautions against possible betrayal, and since the Eldar don't infest a planet with spores they'll have slightly better odds of recovering. However ST ground forces and not up to par and the Eldar are going to do tremendous damage in their inevitable sneak attack, though I doubt they'll be able to wipe the Federation out.
The Tau will likely be overjoyed to see somebody actually accepting their overtures of peace for the first time. Depending on how easily the Tau and the Federation can reinforce each other (I'm not familiar enough with Tau travel times to know if they're likely to encounter each other in a timely fashion) they might try to form an allied block, possibly something like the Federation and Klingons have going on in DS9. This is probably the best hope they have but it relies on a lot of dominoes falling in the right positions initially.
I'll note that as of
Star Trek: Prodigy they have ships capable of jumping 4000 lightyears in a few seconds. This gives them an immense speed advantage and a lot of tactical flexibility compared to everybody else. Sadly they're likely not to be open to tech-sharing.
The Tyranids are a threat but also exactly the kind of threat the Federation is good at shutting down. The Federation
loves enemies that have single-lynchpin members that they can hit and disable an entire swarm. Due to how slowly Tyranids travel, the Federation is likely to find out about them well before they encounter them on-planet and will want to destroy them in space, I'd expect heavy use of cloaked planet-killer weapons such as Tricobalt devices to take out a swarm's Norn Queens in an ambush when it's still months away from hitting their worlds, effectively neutering it well ahead of time and then picking off more chunks of the swarm over a long period of high-speed space-guerilla-warfare. On the ground they'll fare really, really poorly due to Federation Infantry not being
exactly the pinnacle of skill, though I'd expect a lot of attempts at taking out the Synapse Creatures at range in hopes of breaking up the swarm into easily-defeated disorganized individuals.
The real kicker of course, is the Imperium.
"We meet the right sort, this will work. We get some... buckaroo-" Captain Ramius,
The Hunt for Red October
The trouble is the Imperium has too many variables in who they might meet first. If their first encounter is a more rabid stereotypical commissar/tech-priest who wants to immediately burn down anything different things will go really badly. However, the view that the Imperium are mindless savages that immediately open fire on anybody that doesn't perfectly fit their worldview is as much a baseless stereotype as Agent23's characterization of the Federation.
If they meet a more reasonable type they might be able to open negotiations for a time, though the Imperium is always going to be working towards trying to absorb the Federation, even for the Imperium thousands of planets with trillions of humans and abhumans with advanced technology are a real prize. Most Imperium officials will want to take as much of that intact as they can, it's a matter of how much damage they're willing to do to get it
faster. They're going to really, really want FTL that's faster than theirs, doesn't risk accidentally taking a thousand years to jump, and doesn't interact with the Immaterium. Federation ground weapons aren't going to impress but stuff like the Replicators are probably going to.
On the Federation side Space Marines are likely to twig their "Augments, Evil Yucky Bad" reflex and will hurt negotiations with the Imperium from their end.