Before I get started, I just wanted to say I really enjoy the game. I've played it enough to get a feel for what I believe could be even better.
A (more or less) complete wish list for myself:
AI fixes
-decrease how "chicken" the AI is. It runs from practically everything, which might make immediate sense, but means that all you need to do is show up at a planet, spook off the AI, and then fortify your new planet. Move on to the next planet when you're ready and repeat the process. At some point in there, the AI will have to make a stand, but by then you've got such a huge lead in resources, you can usually swamp them. Considering that you're fighting on the AI's home turf, an equally sized fleet has the advantage over its aggressor. Even if the AI is slightly outmatched, it can get reinforcements to the battlezone more quickly than the aggressor usually can. There are a lot of good reasons for it to stay and fight.
-increase the AI's unit diversity. The AI tends to spam light frigates and missile frigates like crazy. This is all right in the early game, but in the late game, it can be pretty easily beaten by the right fleet composition. Advent fleets tend to spam carriers (which is good) and iconus guardians (which is not so good past a certain amount). Similarly, TEC fleets tend to spam Robotics Cruisers, which means its puny little ships are always well-repaired, but those ships are usually overwhelmed by a fleet with more muscle. Vasari fleets never really seem to get past the skirmisher/assailant/destructor phase.
-on a similar note, the AI needs to tech more aggressively to provide itself with more options and more advantages over the player.
-the AI also needs to be more apt with its building choices. Often, one finds the AI does not max out logistics and tactical slots on its planets even in the late game. While this is a problem in and of itself, it means the AI can't prioritize planets like a human can. It's uncommon to find AI "garrison planets" loaded down with defenses and ship-building facilities or AI "finance planets" with trade ports/refineries/research buildings. It tends to get a bit of everything everywhere, which is inefficient at the very least. Obviously, this would be extremely difficult to change - it's hard for an AI to get the "overall impression" of a situation as well as a human.
-The AI needs to know how to capture resource extractors and how to take them from others. Those extractors are extremely valuable, as they're not affected by loyalty percentages and the like and cost little to gain. Currently the AI is extremely clumsy at gaining them or utilizing them, and seems to only take them from the player as if by accident.
Interface Issues
I would love to have the damage and armor type of each ship listed in its description. Why not have a flak frigate changed to say "very light damage 21" and "heavy armor 4" or something to that effect? It lets the player immediately see that this ship is good against lightly-armored targets and is itself a heavy armor ship. It's be nice to see these stats on individual fighter squadrons as well without having to zoom up extremely close.
Ships that do bombing damage need to have that bombing damage kept visible at all times. Now the only time you can see it is when you scroll over ships in the frigate build selection - the built ship does not show how much damage it does when bombing planets.
I would love to be able to see the status of various factions between one another. It's impossible to know if my enemies are allied, in a cease-fire, or viciously warring amongst themselves until suddenly a huge mixed fleet arrives on my doorstep (or doesn't). I think diplomacy could use a bit of work too (give us resources or we'll hate you!), but that's obviously going to take a bit more work.
Balance issues
Planet-bombing frigates need to be re-examined. Currently, they've got a huge supply cost and high resource cost for a ship that effectively exists only to annoy an opponent. Large numbers of siege frigates can blitz defenses in order to wipe out planets, but the resource cost to resettle the planet and the taxes lost usually do not outweigh the cost of the dozen or so siege frigates required to pull off this move, and a lot of those frigates die in the process of going after any world that's even half-way defended due to their light armor and low hull points. Furthermore, they're effectively useless in fleet-on-fleet engagements, doing low damage to other ships and not being able to take a lot themselves. Most players (myself included) would much rather use the considerable supply cost we would've spent on siege frigates buying other ships that help our fleets maintain space dominance over the planet while we wait for our capital ships to take out the planet. You can get one capital ship for the supply of about 4 siege frigates. That capital ship only bombards as well as about 2.5 siege frigates, but it does so many other useful things that there's really not even a good comparison. Many capital ships have upgrades that let them do much more damage to planets than four siege frigates could ever hope to do (provoke hysteria, drain planet, etc.). My personal suggestion is that there's plenty of in-game room for a frigate that does a decent amount of anti-heavy damage. Siege frigates and their ilk already do anti-heavy damage, but so little that it doesn't matter - raising their anti-heavy damage while taking away or minimizing their bombardment damage would give them a role. With an anti-heavy frigate, suddenly there are effective counters for cruisers of all types and enemy planet defenses. Obviously their resource and supply cost would need to be re-examined(hopefully dropped) with such a solution and perhaps capital ships would need to do a little more bombardment damage. Give it some thought, please.
Long-range missile frigates obviously need a bit of work. They're currently the most "spammable" unit in the game because their damage output is quite high for relatively low costs and they effectively counter just about every other type of frigate, which is all most people have in the early game. Whether they need a damage decrease or a supply/resource cost increase is up to you. Obviously you already know this is a problem, as plenty of people have brought this up already. Whatever solution you take, I would just love to see the end result as basic light frigates making up the backbone of a good fleet with LRM-type frigates being viable and useful support. Currently, the LRM frigates can easily provide the backbone for the fleet and that feels wrong.
In an ideal world, with equal resources on each side, it should work this way:
Basic combat frigates > Just LRMs
Basic combat frigates + LRMs > Just basic combat frigates
Basic combat frigates + LRMS > Just LRMs
Meaning the relationship would work out a lot like modern-day infantry with supporting artillery. Infantry will easily beat out artillery positions by just rushing them, but infantry supported by artillery is superior to just infantry or just artillery.
Earlier vasari ships (the skirmisher, most notably) seem a little overcosted. I abide by your judgment on this matter, but it merits mentioning.
Returning Armada is flat-out broken. You obviously already know about this, as well. Because it's such a high-tier tech, it deserves to be good. But right now it's just way too good. Phase Stabilizer gates are already amazing without them generating free ships constantly. I might recommend having it up the supply amount or have it give phase stabilizer gates the ability to recruit ships or something else. As it is now, it's just too good.
Edit: I think it'd be awesome to let player-bought fleets stay "inside" of your phase gates, effectively outside of the game. When the need presents itself, they could be deployed from any friendly phase gate, allowing the phase gates to go from "quick" response time to "instantaneous." That'd be awesome.
Super-weapons are fun, but the Novalith's ability to two-team any planet and wipe it off the map seems a little too good. Why not simply doing 50% of the planet's remaining health? Two-teaming would still be good (taking out 75% of the planet's health), but incapable of doing the job a fleet should be doing. A TEC player can currently wipe out opponents he's never even met in combat by simply wiping out every single one of their planets from afar simultaneously.
The Vasari colonization techs are good (as they give bonuses to two planet types that aren't as good for humans at the expense of better desert colonization), but the volcanic bonuses need to go higher. Volcanic worlds are already fairly low-population and they have a worrying tendency to have unique features that further lower max population. Even if each point gave +25% (for a total of +100% at the end) bonus to population limit on volcanic worlds, they'd still be less profitable than ice planets are right off the bat.
Exploring planets needs to be more worthwhile. It usually finds nothing, rarely finds an artifact, and the remaining planet features are a mixed bag.
Phase Jump Inhibitors aren't much good, as you already know. I don't see what's wrong with them disallowing phase jumps within the system so long as they're up (gives an even bigger reason to have anti-heavy frigates for taking out defenses), but something needs to be done with them. Considering how low their functionality is anyways (in theory they're only useful against opponents who attack your system and then find they bit off more than they can chew and wish to retreat), they need to do more. If they could be made portable (expansion pack idea, right there) and put on a cruiser or something, I'd love them to death (try running away now, losers!), but as it is, they're no good.
Turrets need some work, too. Their attack range is awfully small, which makes it easy to side-step them. I don't want them covering the entire gravity well, but they really need a longer range to be as useful as hanger defenses. This would further give a good reason to bring along the kind of anti-heavy frigates I mentioned earlier in order to rip down enemy defenses (true "siege" frigates, as it were) before assaulting a planet.
I'd love to go on and on about capital ship abilities, but it's been covered before by many others. Some are extremely powerful (shield regeneration on the Progenitor Mothership, for instance) while others seem pointless (the Marauder's subversion ability, for instance, is quite weak). I'm sure you'll be rebalancing these in the next patch.