PS4 Inside – What is the Controller Scheme in PS4 Inside?

Yes, you can use your Vita as a PS3 controller! Go to the remote play tab on your Vita, and there will be a second tab (it's blue). With the new update on the Vita, it allows you to turn your Vita into a controller. I haven't tried it to see how you connect it, but it was just recently added. If you can't see it, update your PS Vita.

There is no way to do it (at least for now). You are required to be online to view trophies. Quote:

The PlayStation 4 will not allow users to look at their Trophies or Friends lists when the console is offline, Polygon has learned using a retail unit received for review purposes.

The PS4 still doesn't sync Trophies in the background — every time you load up your list of virtual accomplishments, the console takes a moment to sync with Sony's servers. Perhaps because of that, you can't view your Trophies list if you're not connected to the internet. On PS3, you can scroll through your Trophies whenever you want, and you can sync them the next time you're online. You can definitely earn Trophies while playing PS4 games offline, but if you try to look at your list when you've disabled the console's internet connection, you'll get the following error message: "[Connect to the Internet] is turned off. (NW-31484-0)"