Its fine! succulents grown dry will lose old lower leaves to provide for new growth. If you want you can repot it, it is probly rootbound, so it might help but only go up by a third to a half or it will be easyer to overwater. what i would do instead is sink it in the ground halfway or in a bed of sand, that will help it stay cool and get all the water it wants
as to the pup, wait untill there is a inch or so of stem, cut with clean sharp knife, dry out in shade for up to 3 weeks anything over a day will do for small cuttings.
you can just chop it and it will ( likely ) grow new smaller plants out of base. let cutting dry before repoting
i hope that was of some use, have a great day