Remote Play Question (PS4 to Vita)

So I have had a Vita for about a year and a half and a PS4 for about 5 months and have yet to try out the remote play option. My question is, after I set up the remote play feature, can I be at a friend's house, connect to their wifi on my Vita and start playing my PS4, which is at home on my home's wifi network? Taking it a step further, could I then play something like Killzone Shadowfall or Battlefield 4 online through my Vita?

I don't have a big interest in remote play if it only means being able to play PS4 games through my Vita in my house since I have a TV dedicated to my PS4/PS3.

Yes you can but the controls are a little weird, for me at least but it's possible.

Good point, I didn't think of that, but a little KZ online here and there when I'm not home can't be a bad thing