I'm beelsebob, the lead developer on Battle Dungeon at Hunted Cow Studios. I can confirm that asynchronous play is indeed the primary method of play in Battle Dungeon. You can choose turn time limits of 10 minutes (if you want a quick game), 30 minutes (so that you can bounce back and forth with another player all day) or 25 hours (so that you've got a day, and then time to play your turn). You can play in up to 20 games at one time, so you shouldn't be short of things to do!
You can indeed play the AI in a ranked game (the AI even has its own position in the leader board, and we'll be introducing more AIs that play by different strategies, and seeing how you all fare against them). We do indeed use the ELO system for our rankings.
I bought the game and it seems like an excellent title with a lot of care going in to the development.
I ran in to what might be a bug, unless the dev can shed some light.
I had a soldier with 8ishAP left. He used one ability, but had enough AP for a further move according to the tiles. When I tried to move him the error...
"Your soldier appears to be in the wrong place"
I had to end my turn, and went on to lose my cleric and the game. No biggie, but is this as intended, or is it an issue?
That certainly sounds like a bug. We'll get looking into it! For now, if you see this error, log out, and back in again (easiest way to do that is to restart the app, as then it remembers your password). This should place your soldier back where the server thinks it should be.
Will there be an update to add the ability to chat with the other player?
Chat is something we've thought about a few times, but have not yet implemented for a couple of reasons (partly to do with managing the chat effectively, and partly to do with fitting it into the game's user interface in a way that doesn't take away from the actual game play). I'm not completely ruling it out, but at the moment, we haven't come up with a really good way to do it, so we're leaving it.