Ps4 – How to transfer trophies from the same game between consoles

Yes. Memory cards (and maybe their data) are tied to Sony Entertainment Network accounts.

If both Vita are signed up with the same SEN account (tied to the same PSN ID), then you can use the memory card without having to do anything.

If either Vita is not signed up with an SEN account, then you can't earn trophies on it, etc.

If the second Vita is signed up with a different SEN than the first, then the memory card usage will be restricted. And to use the memory card, you'd have to format the card.

While not clearly outlined in this source, I suspect PC/PS3 transfer would have similar restrictions although I didn't look into it.

Source: Playstation

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)"