I'm very new to GalCiv games. But I did play MooII a lot, and have some thoughts for improvements based off that game.
As my sub-title suggests, this game is a awork in progress, and it could become great game after a few more patches.
I'll give some random bad advice based off what I've seen so far. Other ppl might correct me where I'm wrong, as I will be, and comment where do not agree, which they won't.
All this after the 1.1

1) Campain.
After learning the game I played a campaign. I set difficulty to tough - since a forum poster complained about "normal" being too easy. I notice I'm at war with Drengin, so I hurriedly start research into space weapons. Build some quick fighters - and then spend 40 turns killing off unarmed enemy ships. FInal military report: Killed off 57 enemy ships, 0 losses. Not due to any skill of mine, but due to the fact that I never fought an armed ship. Totally laughable performance, and not good PR for this game.
Fix: Let computer design ship more frequently at the beginning of games?. MAKE it so that the Drengin will have some ships at start with basic weapons. I didn't feel like continuing the campaign after that, waiting for the patch.

2) Where are those ships again?
After 2,5 full games I find myself playing 97,5% of the time in zoomed-out mode (the one where you see brightly colored ship-symbols). Why? Well it's easy to see the planets, and the playing area felt a bit small when zoomed close (too much map-dragging) but most importantly - I couldn't find my/enemy ships! Kept missing invading ships, and forgot where I parked my own. Why? Well, call me short-sighted or whatever, but my ships quite dark texture, and space being black and all - I simply couldn't find ships. Same with anomalies.
Fix: Option: Find a way to mark ships/fleets/anomalies so they stand out a bit. Color might be a hint here, I can see all ships very plainly on the "big picture" zoom level.

3) El AI stupido.
The "AI declares on a vastly powerful enemy" is already mentioned, and did happen in my most recent game.
The game before that I had a vastly powerful runaway AI that outclassed me in every way. It declared, let me build up for 8 turns or so, then started sending unescorted transports into my systems. It eventually used some escorting +6 beam attack 10 hp fighters (which I could defeat with fleets of 10 hp +4 attack fighters). I noticed that each & every planet had +26 attack medium ships sitting prettily on display.
A short comparison to Moo2 - the AI declaration of war was followed immediately by the dreaded (Klackon fleet attacks the Sol system) message - no time to prepare now. See the signs or die.
About embassies - that game I lost had the AI building embassies like usual, and it did push my planets into rebellion. So it's not always stupid. But I bet he built them on core-worlds too...

4) Combat system.
Already discussed elsewhere. I would agree ship initiative seems like a better way to go. Ships have initiative (based off speed), but also with a random factor. This will make battles more fair on the AI (who is too stupid to realize he has to attack), and it will make them a bit less predictable (if that battleships fires before mine I'm toast) etc. Either calculate one initiative order and stick to that through the battle, or work it out every round for more chaos.
Moo2 put initiative in patch 1.31, after the 1.2 patch. So there will be a nice symmetry here. AI needs not be reworked, it could not possibly fare any worse than it does now.

5) Ship upgrading.
Correct me if I'm worng. I couldn't find a way to upgrade a single ship to something, or "add" just one module later. Is there a way? Should there be?

6) Ship stats.
I would like an easy way to access a ship-class' stats. Showable in-game as well as the shipyard.
Like Hull - Engines - Weapons - Defense - Modules. (Tiny - 3 x laser II - 2 x warp II - n/a)
Change the output when you add stuff.

7) Ship design.
My favourite part so far. Even though I play zoomed out (see 2) I spend some time here. X-wings, Y-wings etc etc. Loads of fun. But even so, how about an optional quick-design feature? Like Moo2, you just click "add laser" button etc.

8) Ship upgrade/construct.
I must be stupid. Is the a way to get to the upgrade/construct/guard menu when 2 ships or fleets are in the same parsec? I had 2 constructors in same square, had to move one out, then click "construct" with other, then move the first back in for add-on. Same with upgrades and ships.

9) Tech paths mutually exclusive - idea. Something from internet-based "planetarion". Make it so that certain (near end) tech excludes you from getting certain other techs. Say, you research majesty. Well, now you can't reseach (say) Expert trading or something like that. Give ppl some choices, not just "reasearch all then win". 10) Good luck on further development. This could turn into a great game.

(Btw. - I played on a quite hard level (crippling) or something for my 2. and 3. game. So it's not stupid AI by setting).