Played through 400+ turns as the Dominion, some observations that I hope will be helpful! I was on the 11/15 release, so it is possible that some of this is already fixed in the 11/21 release.
1. I basically cruised through the universe. Roms put up some resistance, and the Borg were fun defeating, but otherwise, not so tough. I'm not certain the AI ever built troop transports. I never conquered a planet with a different race than the major power I defeated, so the AI's never took systems from one another, and I never saw any troop transports during my conquests
2. Head to head, the Roms generally beat me in ship battles. For example, I had 10 level II and III battlecruisers, going up against 3 cloaked Level I and a single level II battlecruiser. Each time I went up against this group, I'd lose 2 of my ships and they'd retreat unscathed. I finally dropped 20 level IIIs on this group to finish it off. Not sure if cloaking should be this powerful? For the first 1/2 or more of the game, the Roms were ahead of me in Tech Level.
3. As I advanced on the Roms, they gathered 44 ships into a single fleet and parked it on Romulus. I could only support 15-20 warships total, so 2 detachments of 10 ships each from this stack could easily have pushed me back.
4. I started building an Outpost base at a formerly Romulan system. This got the AI riled up. It sent all 44 to destroy it, allowing me to swoop in behind it and conquer Romulus. Later in the game, this 44-ship fleet was parked at another (new capital) system and building an outpost nearby got it to move again.
5. I found that when I raided a system to destroy the shipyard, the AI never rebuilt the dock, making it much harder for it to counter attack. I never did take down that 44-ship fleet, just stranded it in a single system. I never brought my ships near it, so it never ended up attacking me.
6. Question: How could the Roms support a 44-ship fleet with only 3-4 systems and miserable morale? Answer: they have access to an unlimited line of credit! Interest free, I'm sure!
On the Intel screen, it shows them with a treasury balance of negative 550,000 credits! (Klingons have -212K). For my part, I discovered why I need that building for scrapping ships. 
7. I struggled for (almost) the entire game with morale. I couldn't seem to be able to stop morale falling on conquered systems. Spent a lot of time "culling" the population. Membership Minors kept decreasing too (I didn't subjugate any minors), which was more frustrating. Not saying this is all bad, necessarily. Doms need a special that helps Morale system wide, I think. Had the AI played differently, I wouldn't have been able to sustain all the subjugated systems. I lost systems to rebels twice. Morale issues for me were worse as the Roms fared worse at my hands. I guess their starting morale, post subjugation, was lower because Rom morale overall was lower due to their loses. The rebellions were on two of the last systems I conquered.
8. At one point, the Borg assimilated a system of mine, leaving a small fleet of mine stranded. I think every system got a morale hit, every turn, while it was stranded? Took a while to get it unstuck. I found myself wishing the Borg would destroy it, but no luck.
9. I don't think the Borg or Cards expanded very well. Feds spawned near me, and I finished them off early. Cards were stuck between Roms and Feds/me, so that didn't help them, I'm sure.
10. Unless the Klingons ended up fighting a lot with the Borg, I don't think the Borg did much either. They only assimilated 3 systems (once minors, I'm guessing) before I met the Borg. Once I did, they again seemed to bee-line to my systems, leaving their home system unprotected. They weren't much of a threat though. 1 fleet of 2 cubes/1 diamond, and a second fleet with a single probe.
11. I ended up with Membership treaties with every single minor on the map but one (Mizars) which was deep in Klingon space and likely surrendered immediately upon meeting them. I pretty much defeated the Feds before they could connect with the minors, so they never factored into the competition. Neither Cards nor Roms managed to secure a minor race.
12. I hadn't yet started my war with the Roms when I made first contact with the Klingons. I think Klingons were allied with the Roms though, as when they declared war on me, their allies, the Roms also did. At that moment, I had subdued the Feds and was building up to finish Cardassia. I wondered if could sustain a war against all the majors at once, which was kinda cool. Ultimately, the Klingons never sent ships into my space though, and the Roms "turtled" their fleet so I could peel off their systems 1 by 1. Much later the Klingons offered non-aggression, allowing me to focus on the Borg.
13. Borg was an interesting cat-and-mouse battle, and my best asset were... colony ships! I'd terraform a planet, colonize and build a shipyard. The one Borg fleet would then charge and assimilate the system while my fleet hopped the next (empty) system and did it again, getting closer to Unimatrix 01. As soon as the Borg left the first system to attack the second, a second fleet of warships/colony ships would destroy the first (now assimilated) colony and then recolonize it behind the Borg. Once I nailed Unimatrix, the Borg were stranded.
I'm still learning aspects of the game, so I very much enjoyed the playthrough. In retrospect, it seems like the outcome was inevitable though. The AI needs to do a few things differently:
1. expand a little better? Rom and Klingon did colonize a sizable number of systems, but Feds, Cards never had more than 3-4 systems. it was a small galaxy map, though, so perhaps they just didn't have the space?
2. build troop transports - and use them.
3. win over (or conquer) nearby minors. 200+ turns into the game, the minors closest to Romulus shouldn't still be independent.
4. be more aggressive with their fleets, split them up and use some to protect outer systems and counter attack. don't turtle. don't overprioritize targets at the expense of other objectives
5. rebuild lost shipyards and prioritize them for growing outwards
I'm sure all of the above is more easily said than coded, but I'm hoping this helps!!
Some of the things that went well in the game include:
1. the Rom/Klingon alliance
2. AI tech levels. I thought my Tech Level was growing well, yet Roms where ahead of me for much of the game. I never conquered a Klingon system, so I don't know for sure their level. they were bigger than the Roms, so I'd guess they were ahead of me as well
3. Borg fleet was still very dangerous, but it wasn't a doomsday machine heading straight for my homesystem like in earlier games
4. Rebellion! Conquered systems weren't cooperative. they needed to be managed.
5. Not having to start over at level 1 in new colonies!!!! In my previous games, i had to build level 1 buildings in new colonies, and upgrade them repeatedly. having them start at my current tech level eliminated a lot of tedium!
Thanks for making this game so enjoyable! Next game I'll start at highest tech level, so I can see what the higher levels are like.
they always seem to get the crappiest starting position, with few nearby stars, and minor races in the vicinity - which hampers their expansion. But it's not an issue specific to the Cards, AFAICT. The code is pretty much the same as for other empires.
