First, your 3 questions.
1) It isn't an *impossible* task to take the homeworlds now, but now with what you have *right now*. Assault Transports and warheads can make it possible, though your main firepower should be Spirecraft.
2) Spirecraft exowaves increase with both time and AIP. You're almost 24 hours into the game, so the exowaves are probably near the upper end, but I don't think they're at max yet.
3) The nebula you are describing is one of the harder ones, especially for where you are. You can try the other nebulae to see if a different random one will spawn, but if that doesn't work, just lose it and go on. The Champion at destroyer level is only middling useful. It's a powerful, and free, unit - but it's basically a good starship unit at this point. On the otherhand, a Battleship Champion is more like a Hunter/Killer superweapon. Of course, right now the AI's Nemesis response is also pretty insignificant. If you do go on, be aware that if you finish the nebulae line, the AI will respond with something like 8 times the response it does now.
General notes:
Looking around, I strong recommend getting all the AIP reducers first, and doing it early. If you were to clear out the Coprocessors, data centers, and ride the SuperTerminal for about 40-60 net AIP reduction, you'd be at AIP 300 with an AIP floor of 200.
Then, both of the Spire Archives are still around. They're in inconvenient spots, yes, but try clearing but not capturing both systems, then use your whole fleet to protect one while building heavy defenses and protecting the archive. Once it is fully drained, abandon the system and move to the other archive to repeat.
Doing both of these would drop 100 AIP and net roughly 30,000 Knowledge.
Then, there is also still an ARS available. Unless all three ship types are useless, it is worth picking up.
I recommend capturing or hacking more Mk V turrets - They'll help you with your exowave problems. Getting the Sniper Mk V is an excellent first step - Sniper and Spider are probably the best, with Laser not far behind.
However, looking back. You split your Knowledge a bit too widely. You purchased Harvesters Mk II but not Mk III, Logistics Mk II but not Mk III, and Econ Mk II but not Mk III. Splitting your K like this is usually inefficient. Harvesters require a large number of resource points before they become worth it, and even then they lack the benefits of the Command Station lines. For you, upgrading to Mk II Harvesters gained you +800 M/s or so. But upgrading to Mk III Econ stations would have gotten you +1536 M/s and + 840,000 Energy instead.
On the other hand, Logistics Stations are, like Military Command Stations, a combat multiplier. Their useful effect is reducing the speed of enemy units, giving you more time to shoot at them. Upgrading the Logistics Mk II to Mk III would have gained you +800 M/s, but also about 20% additional reduction in enemy speed.
Personally, I prefer Military command stations, especially Mk III, for dealing with CPAs and Exowaves. The +100% attack boost applies to all your turrets, units, and forts, instantly doubling their effect.
You've invested in starships, but not too heavily. Mk IV bomber is an excellent choice, almost required, but fighters is less useful. Harvester Exoshields is something I would recommend *never* getting.
As Brokentoy pointed out, you have a lot of high-cap low-HP fleetships like Raiders and Raptors. These are weak for the HW assault phase. On the other hand, you have unlocked the Zenith Devastators somewhere along the line. These are very high-power forcefield ignoring units that are perfect for taking out AI starships, Core Guardposts, exowaves, and such, especially when the target is under a forcefield (Exowave Shieldbearers, I'm looking at you). Unlocking Mk II-IV Zevas will be a major boost to your attack power.
Assault Transports actually go well with your high-cap units, as you can fill them up without losing much external firepower.
Warheads are a major force multiplier, although the AIP price is painful. Fortunately, when you are at the AI Homeworld Assault phase, you don't care too much about AIP, so... go wild.
And Spirecraft. Spirecraft are superweapons for a reason. Cap up on Shieldbearers, Martyrs, Penetrators, Implosions, and Siege Towers. Send them all in with your fleet, and watch the AI melt away. Spirecraft Transports are a cheap (as in, almost cheating) way to get units close to distant targets (such as the Wrath Lance on Oorisini). The Martyrs will basically remove the Strategic Reserve from play, the Penetrators can quickly pop the most dangerous targets (CPA or CRE guardposts), while the Implosions and Siege Towers are just general high firepower units.
I you focus on building up your Spirecraft fleet, you could probably take both homeworlds without doing anything else.