Regardless of what you have open, or if you don't have anything open at all, it should streetpass without problems Otherwise, you could only get streetpasses in one game at a time.
Though, I've noticed that most people have deactivated animal crossing for streetpass. Paranoia about getting random move-ins, I guess
As long as you're connected to the internet and have your streetpass activated, you can receive streetpass whether you're in sleep mode or have another game in c:
Not for Streetpass. You need to have it activated for ACNL then just take it out with you. If you Streetpass someone else who plays, you will get the information for all of the games that you have in common, as long as you both have the capable games activated in your Streetpass information.
Confirming that you don't need an internet connection for streetpass to work. My 3ds is never connected to the internet when I'm out and about and I streetpass just fine Well, I would like more streetpasses, but I get them often enough to be sure that it works.
Just in case you don't know, you also have to have the Happy Home Showcase and I think you have to talk to Digby to set it up. After that, put the 3DS in sleep mode and go to a Home Depot. Seriously, Home Depot is a streetpass oasis.