I have a slingbox PRO-HD & a slingbox 350 at separate locations. I've used both the iphone 5 slingbox app and the ipad air slingbox app to airplay both streams to my atv3.
Specifically to your question, backgrounding is seamless. Multiple times, I airplayed World Cup games this summer from my ipad, while simultaneously reading, looking up stats, etc. From both the iphone and the ipad, the picture is very crisp. Buffering is an issue, but that has more to do with network speeds, up & down.
However, the model of the slingbox you use, will affect your airplay capabilities. I have no problems with the 350 stream being airplayed from either the iphone or ipad. The PRO-HD stream is another story. The iphone stretches the screen horizontally(takes a 16:9 image and stretches it even more) and the ipad squishes the screen horizontally (takes a 16:9 image and makes it 4:3).
It kinda drives me nuts having to use multiple remotes/devices to control what I'm watching on my tv, so I don't really use airplay for slingbox. My understanding is that slingplayer for roku operates similarly (ie you need an iphone or ipad in conjunction with the app). So I looked for other connected solutions.
Currently I'm using WDTV Live because it has a slingplayer app built in, no airplay necessary. This allows me to use one remote, Harmony Ultimate, to navigate through the slingplayer app while still controlling the tv, receiver, etc. without picking up another remote/device. I can also program favorite channels on the touchscreen, negating the need to navigate clumsily through the cable providers program guide.
Boxee Box was great as well for streaming, but it had a Wifi/RF remote (can't remember which), which doesn't work with my universal remote. It was definitely the best streamer I've used though.
I've looked into this stuff a ton (probably too much)