1. Mind the bag in my absence.

2. Mind the bag while I am away.
3. Mind the bag when I am away.
4. You should mind the bag in my absence.
5. Did you mind the bag in my absence?
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Can we use all the expressions? Are they all grammatical?

Can we use 'watch' instead of 'mind' here?

They are all fine. In the U.S. we would be more likely to say "watch the bag"; in the U.K, "mind the bag". They both mean the same thing. Both mean, "Pay attention and don't let anyone steal the bag."