Ive got a partial answer for you. Check out iFunBox. Lets you alter the multitasking settings for each app. You want to Allow Background or Always Background for those apps you that you want to behave accordingly. For instance, I have a game that when you close out, always starts at the beginning. When I set it to Allow Background, i can close out and then come back in and it picks up where it left off.
Caveat is I am not sure changing the setting works for every app and I think depending on how the app is built, it will react differently. For example, if I set my Gmail app to Always background but close out of it, it will not refresh in the background. You still have to go into it and then it pulls the email. Also, video apps do not seem to work - meaning you cant keep the audio from a video app like Slingplayer running let alone the video.
But this def works and combined with InfinityTask may work nicely.
See article here. Test away.
http://www.idownloadblog.com/2013/01/24/ifunbox-powersavingmode/#disqus_thread
PS. I think this can only be done on windows version - at least i could only do it that way.