App that survives a factory reset
Your chances for #2 are pretty good – though there might be some caveats. To figure which those are, let's take a look of what a factory-reset does. It...


wipes /data
wipes /cache
wipes the dalvik cache (usually implicitly, as in most cases this resides in /data; I guess with Lollipop it will do the same for ART).


So you see, /system remains untouched. With your app installed in /system/app, it will survice a factory-reset. But its data might not, as /data gets wiped. Which means, all required settings need to be part of the app itself, or of its "default settings" which get deployed on first initalization.

For the second part: That's better asked at a development site (which we are not). Basically, no app gets root access by default AFAIK. They get access to everything their permissions allows (and yes, there are permissions which can be granted to system apps which are not available to "user-space apps") – but for root access, you've got to ask su, which should trigger the corresponding superuser app.

PS: As for the SuperSU popup, you will have to live with that for the first call, I'm afraid.