I'm Looking for some great advice to move forward and make this first experience as smooth as possible. 

If it's a standard Lakewood up/down duplex it probably has a plaster ceiling under the ceiling tile-- at least all the ones I visited when my son was apartment hunting were plaster.IMO, the "nicest" fix (assuming the old plaster is in poor condition) is to scrape off the acoustic  tiles and then drywall over the old plaster with a smooth finish; the cheapest fix, if the only problem is the one tile you pictured, will be to screw up the tile(s) that is/are lose, cover the screw with a little joint compound and then primer and paint the ceiling; If you are lucky the tiles have already been painted before.

If you feel you must worry, worry instead that your all-cash offers are as low as possible (because you'd be paying cash), and, that your juggling with your relatives' HELOCs will proceed smoothly, in a timely fashion.

Lay it flat and put a bunch of dowels between them then when it's lined up, start pulling out the dowels to make contact, and roll it down with a heavy roller (or a can of something) to keep it smooth and keep pulling out the dowels as you go. 

Whenever you are buying a property out of state you should do a few things to ensure it's as smooth as possible.

By the way, the more favorable lender was able to close almost a week earlier than anticipated.Everything went pretty smoothly from there.

Have been involved in multiple 1031s either as a CPA or an investor with him and it always goes smoothly.

By investing conservatively, in good middle class neighborhoods we never suffered much in losses and by not having loans our flexibility and ease of purchase (cash) and sales (seller financed) always went smoothly.

All tenants moved in May 1st and overall has been going smoothly sense.

So I try to keep things as smooth as possible.