I've actually played quite a few of these games. Here are some quick rundowns if it's still relevant:
A couple overall comments: First, I think the fundamental strength of Dominion is that the game is about trying to get the right cards into your hand each turn, and the way you do this is build your deck. All of the cards either do something good (Gold, Sea Hag), or work to change the way you draw cards (Village, Lab, Smithy), or both. Most of the first-generation clones had the cards do "something exciting" . But it turns out that if you combine deckbuilding with cards that don't change your deck or hand, you're just creating a slow action-drafting game and the fact that you have five different cards each turn doesn't matter.
Second, for whatever reason every designer's first instinct when messing with Dominion is to add a second currency (and usually call it "fighting"). In theory, I guess this could be interesting. In practice I haven't seen a game yet where it added much. (I'm still not enamored of the Potion-cost cards in Dominion, either.) But it lets you have a thing called "fighting" and there are definitely consumers who respond well to that, judging based on what Kickstarters succeed.
Thunderstone: I played this once and liked it a lot. I played it a few more times and liked it noticably less each time. It takes way too long and the interactions between the cards are very blunt. The first edition also had weird card layout and questionable balance. They released an entire overhauled game, Thunderstone Advance, which is better, but has the same fundamental weaknesses and definitely still takes too long on the table. Remember in the introduction where I complained about not being able to do much to make the cards in your hand work together? Well, Thunderstone Advance came up with a Band-Aid rule that lets you spend your turn to discard as much as you want and draw up to your hand size. It's better than nothing, I guess.
Arctic Scavengers: The idea of having every card be able to do one of several things to varying abilities is a neat idea, but I don't think the pieces work well together. One of the actions is called "fighting," which gets people excited. They would be much less excited about the mechanism if you called it what it is, which is a "blind bid". There's a second currency, medicine, but you can't just buy it; you have to get lucky digging through the "junk pile". Yes, imagine if half the cards in Dominion had Potion costs and the only way to get Potion was through the Black Market. Luckily there are cards that let you look at more Black Market cards!
Nightfall: OK, it's Dominion with directed attacks and the goal is to get the most Curses into the other players' decks. Sound good? You can only play one action a turn, except that there's this arbitrary "chaining" mechanism that allows you to play more as long as you can match the colors of cards played by you or the player previous in turn order. The next time someone complains about Dominion's theme, ask them what the hell the colored moons in Nightfall have to do with anything.
Despite my dislike of the game, Nightfall had a really interesting deckbuilding innovation that I admire. You get an initial deck of low-powered cards, which span most of the game mechanisms in an easily understood way, and trash themselves when you play them. This helps novices get a grasp of the game easily, and provides a built-in transition from "starting deck" to "midgame deck" without having to explicitly introduce cards that mess with your deck. I like that. I'm much less impressed by the idea of drafting individual supply piles before the game begins. In theory it leads to different, asymmetric decks during the game; in practice it requires you to make the most important decisions before the game begins.
Ascension: At its heart this is a very basic Dominion clone with two changes: a second currency, which is fighting of course, and buying cards from a rotating, not fixed, supply. I thought the game was thin, but it seems to have a strong following, so I am still waiting for the long-anticipated Android version so I can give it a shot. I think it's definitely best with two players, where your choices to buy or not buy from the center are much more strongly related to your outcome.
Quarriors: I think this is a fine game in that it lives up to what it says on the tin, and what it says on the tin is "The Game of Uber Strategic Hexahedron Monster Combat Mayhem!". The idea of having multiple cards that refer to the same die is brilliant. The game itself has a lot of luck, but you signed up for that when you walked in, and there's a fair amount of strategy in choosing between your different purchasing options, too. I highly recommend the advanced variant in which you can buy two dice a turn and to score a die, you must cull that die--it makes the blunt strategy of "always buy the most expensive thing available" less dominant.
Puzzle Strike: Of the games that could reasonably be called Dominion clones, this one is definitely my favorite. This is because I think it actually keeps the good parts, while introducing interesting innovations. The Action cards (chips) have different colors and many effects that provide +Actions only give you actions of a specific color. There is lots of drawing and Actions with interesting effects, which makes the mix of chips you have very relevant. The primary innovation is the victory condition--as you attack your opponent, you bring them closer to losing and yourself farther away--but the closer you are to losing in this way, the more chips you draw, which lets you either buy big chips or (hopefully) play your big combo. The second innovation is the character chips, which lead to an asymmetric game. For advanced players, the way you play a board will definitely depend on both your and your opponent's characters.
This is not to say there aren't weaknesses. First, the game was clearly designed exclusively as a 2p game. The 3-4p rules just don't work with the "unbalanced seesaw" pacing and victory conditions. The game has been through three editions, with the chips changing in each, and has earned some IMHO justified criticism for not bothering to get it right the first time. (Remember the story about one pile ending Dominion, and the dominant Duchy-rush strategy that some playtester showed off? That's the sort of mistake the pre-nerf Combine was, and good development should have caught it.)
Eminent Domain: This game is its own animal. The deckbuilding and play are completely different than any of the Dominion-like games I've talked about above, and yet the deckbuilding still works and is important. You get only one Action a turn, and almost nothing gives you any more--then you also get a Role, which allows you to play any number of matching cards in your hand to increase the effect. But when you take a Role, everyone else can play the matching effect too! Eminent Domain has a rhythm which takes some getting used to but I really enjoy it.
Mage Knight: The deckbuilding is only a small part of this adventure/exploration game, and I wouldn't really want to talk about it in a discussion of deckbuilding games. I thought the player interaction was pretty minimal, and I found myself wanting to play solo--because the puzzle of a single turn is fascinating, but the downtime between turns was pretty extreme. But it's big, bulky, and fiddly, and what I really want is to play this on PC or tablet.
By the way, Ratsia, thank you for your quick review of Trains. It's been on my radar as something I want to try. I have high hopes, but then I had high hopes for a lot of these games.
Edit: because I do all my best game thoughts in the shower...
Friday: This is a cool solitaire game. You need to line up your cards to pass a series of challenges. When you pass a challenge, you add the challenge card to your deck--turn it upside and it's an in-hand power. The trouble is that your deck starts out very, very bad. When you fail a challenge, you lose HP, but you are allowed to cull the bad cards that helped you fail. I enjoyed the game and admire the design, but like Mage Knight, I want to play solitaire games like this with electronic aid.
Core Worlds: A lot of people like this game, but I'm not one of them. The fact that you see each card only a small number of times limits the interest of the deckbuilding aspect, and the interactions are blunt. (There are Robot cards, cards that give Robot bonuses, and an endgame card that gives you points for having Robots.) The fact that there is a game arc is interesting, but the five different-deck phases make it feel a bit forced. Also, this game is long, in the 90-120 minute range.
Fantastiqa: Of all the games listed here, this is the one I dislike the most strongly. There are not one, not two, not three, but NINE different currencies, all of which are exactly identical. The game is built with this lavish fantasy theme, but swords work the same way as wands work the same way as spiderwebs. You use your currencies to wander around a tiny board picking up cards in a way that doesn't matter. The quests that come up are random, and require you to get 2-3 each of 2-3 different currencies. The way you get these to line up in your hand is, you sit at the appropriate destination and use the action that lets you pass your turn and discard only the cards you don't want until you've drawn everything you need. Some of the art is very attractive and well-chosen public-domain art, which is the game's best redeeming feature. Unfortunately, this is marred by the iconography and other graphic elements, which are awful horrible clip art; this is particularly jarring when juxtaposed against the lavish paintings used elsewhere.