Kaybetmek = to lose something
This is a transitive verb, and it will require accusative case on the object that has been lost.
- Anahtarlarımı kaybettim = I lost my keys
- Kedimizi kaybettik = We lost our cat
- Aklını kaybettin mi? = Have you lost your mind? ;-)
Kaybolmak = to be lost
This is an intransitive verb, and the lost item is now the subject, so it stays in nominative case:
- (Ben) Kayboldum = I got lost
- Kedimiz kayboldu = Our cat was lost
- Anahtarlar kayboldu = The keys were lost /vanished
(This is a good way to deflect responsibility: I didn't loose the keys... The keys got lost on their own, haha!)
I will check with some friends, but I do not think so. If you do that, you would be telling someone to literally get lost (and in to intentionally lose their way).