I'm pretty sure you can choose the order of the non-Avatar units, but I'm not certain. Can anyone confirm one way or the other? I can't check for myself right now.
When you do the download, you get everything that's ever been available downloaded onto your game, but it doesn't show up ingame until you select it from the SpotPass menu. So from that point, you can summon whatever teams you want, up to a total of 10 at one time. Also, you're a bit limited initially in the teams you can summon from whatever the most recent series: when the FE1 team becomes available, for example, you'll first have one character available, then 24 hours after that, you'll get another, and so on. They all become available when the next team comes out, though.