It's a fairly small developer, VERY "indy", so a demo of the game isn't likely to come any time soon as a separate "trimmed down" version of the game would then have to be made just to provide a sampling of it...
The turn-based structure of the game is perhaps a negative to some, but to those who long for a game like MoO2 and the like, the turn based structure is a welcomed and desired feature. Real-time is great for games like Sins of a Solar Empire, who focus on a handful of stars, but games like Eternal Space and MoO2 for instance can easily involve dozens of stars along a "border conflict", particularly if your opponent is on a comparable level of advancement.
The card-system is kind of "gimmicky" in some respects. I definitely see a reason for it, and in some ways I do like it, but I typically just make sure to have a good fleet first, and spam "repairs" so that my fleets leave the battle in good shape. Sometimes, that little bit extra damage or accuracy is handy, but what's better and just plain more reliable is simply having a better fleet.
As to the influence in battle, I am definitely on the fence with this one... I do miss controlling combats like in MoO2 if I want, but in all honesty once I reach a sufficient level of power, I don't want to waste time fiddling around with it, so I would set it to "auto" and watch my ships mop-up the enemy. That is in-effect what Eternal Space is, but all the time... There's no ability to turn that off, so in some ways it takes out the ability to plot out your combat exactly as you'd like.
Is that a game breaker? For me, not really, I first had to learn not to auto-design ships so that they could be more well-rounded. Focusing entirely on one type of defense is nearly suicidal, no matter how much better the ship is, because it gets torn up by crappy ships doing damage it can't defend against very well. Manually choosing better and more defense options is paramount, and the variety is as influential as the amount from my experience.
The weapons don't seem to have too much pro/con issues, so I've had fun piling on whatever I liked. Bonus modules are also darn handy, and the auto-builder often ignores them, meaning you can easily miss out on the extra build-space module(s) which are practically a given. Sometimes I like making "invaders" with decent defense/offense, but simply tons of invasion boosts so that I can conquer systems quickly and easily before much of a reprisal can muster.
All of these things are of course parts of the game I've fiddled with so far, and I'm -REALLY- hoping to have even more features added to the game before launch. Perhaps a software development kit (SDK) as well so that ambitious modders, such as myself, can take things apart and re-work them as we like to get even more out of the game. For now though, the game shows a tremendous amount of potential.