iPad mini Connected to computer, privacy

While there is a chance you could get malware from the computer, it is vanishingly small as it could really only take the form of an infected app.
Trusting the computer means that a backup could be made of your phone in iTunes which could hypothetically have passwords backed up. Since you will need to use iTunes to transfer photos and data, just make sure iTunes doesn't complete a backup. The best way of avoiding this is to turn on iCloud backup.

 

What the previous poster said is correct. But the trickier part is that an iPad can only sync with one computer at a time. So if your iPad is synced with your own computer, syncing with your parents' computer would delete any stuff you synced from your computer.

Much as I hate to say this, the simplest way to get pictures from your parents' computer onto your iPad is to copy them onto a USB stick, bring it home, copy them to your computer and sync from there.

 

No, the trust option doesn't mean access to your passwords. Go ahead and trust it and transfer the data. You could do it without using iTunes if it's just photos and video but the source video would need to be in native iPhone format. Then you could copy it to you DCIM folder without iTunes.

Anyway, avoid the hassle and just do the sync. Be sure to start by disabling auto-sync. Then choose to not sync music, contacts, apps etc. unless you want all of your mom's stuff on your phone (I assume you don't). Finally, run a sync that only does the photo and video.