How do you find the wall studs without removing the plywood?

I tried to install a 15"v by 30" h kitchen cabinet in my trailer this morning. After about 15 #8 1-1/2" were screwed through the back into the side wall, apparently none into the studs, I can say it's attached, but it wobbles if I pull on it and I have absolutely no confidence that it will hold any weight. On top of that, the Home Depot droid suggested I use brass screws "so it will look better". It occurs to me now that brass screws are probably a softer metal which would explain why half the screws stripped while I was installing them... (sigh).
Since I'm going to have to remove the cabinet and reinstall it properly, perhaps you can offer some advice:
1. How do you find the studs? I have a stud finder for wooden studs under drywall -- but that wouldn't work in a trailer, right? And wood drill bits would just spin when they hit the metal stud, true? I'd need a carbide bit.
2. Is there an easy way to remove a stripped screw? I have one of those reverse-grip bit kits that's supposed to be the thing for removing screws, but all they've ever done for me, after hours of work, is to grind the head down until it separates from the rest of the screw.
3. How thick is the side wall? I'm terrified of going through the outside skin. What length screws do I use? (Through the cabinet wall
and its attached ledger is just under 1".)
Thanks in advance. I thought this was going to be so easy...
Edit: Added picture. If you look closely you can see a silver screw in the plywood underneath the cabonet where, presumably, the stud is... But there is nothing similar 16" away...

Since I'm going to have to remove the cabinet and reinstall it properly, perhaps you can offer some advice:
1. How do you find the studs? I have a stud finder for wooden studs under drywall -- but that wouldn't work in a trailer, right? And wood drill bits would just spin when they hit the metal stud, true? I'd need a carbide bit.
2. Is there an easy way to remove a stripped screw? I have one of those reverse-grip bit kits that's supposed to be the thing for removing screws, but all they've ever done for me, after hours of work, is to grind the head down until it separates from the rest of the screw.
3. How thick is the side wall? I'm terrified of going through the outside skin. What length screws do I use? (Through the cabinet wall
and its attached ledger is just under 1".)
Thanks in advance. I thought this was going to be so easy...
Edit: Added picture. If you look closely you can see a silver screw in the plywood underneath the cabonet where, presumably, the stud is... But there is nothing similar 16" away...