No. The token is a IOU receipt to be able to redeem lightning sats from that specific mint.

You have to have the token, and the mint must be online to issue your sats when requested.

The mint "could" rug anyone by not giving the users their sats when asked, just like a bank run

Always keep your mints, always backup your JSON tokens for those mints (minibits is great for downloading backups). Once you know for certain that you "spent" all your ecash notes, and received the sat amount (minus fees, etc.)

Then I'd say go ahead and delete the mint/ecash wallet

Always always always backup your ecash tokens (JSON files)