No you would not say "Can I wear this out?" Because 'to wear out' is a phrasal verb meaning 'to use so much that it is no longer able to do what it is meant to do. 'Worn out clothes have become shabby, misshapen, maybe they have holes in and have reached the end of their life because they have been worn many times. In your example I might say "Can I wear this home?" or "I would like to wear it now, please"

No. 'wear something out' is a phrasal verb that has a couple of other possible meanings, but not right for this situation. You can say "I want to wear this/these now." or "I'd like to put this/these on now/straight away."

You can say 'Can I try this out?'