Are these red-flags or normal practice for accommodation in Georgia (country)?

My answer is not specific for Georgia, but I wouldn't consider your issues a red flag in any country.

Up-front payment is quite common, especially in smaller or private-run accomodations, which only accept cash. Even in larger hotels in western Europe, I occasionally have to pay when checking in, and even if it is uncommon or odd, I don't see why this is considered a red flag? If you are afraid that the lodging is so poor that you won't stay, then this is probably something you have to take up with booking.com anyway instead of expecting an on-site reduction of pay if you decide to leave early or only use a part of your original reservation.

Refusing to give a receipt for any service or purchase is also quite common in all countries with dodgy fiscal systems. The landlord is very unlikely trying to frame you, but to avoid taxation.

Coming from another country, where it is also uncommon to lock doors, at least in rural areas, I would also not find the missing key very peculiar. I am not sure what kind of accomodation you have booked, but from a safety point of view, there are many types of accomodation (hostels, dormitory-style lodging or private room rentals), where it is not common anywhere to get a key for your 'area'.

I just came back from my two week stay in Georgia, so I have some anecdotal evidence (of course, this is not a definitive answer, just some datapoints).

I stayed in these three places:

1. An apartment in Tbilisi ~ a week, booked via booking.com. Pay up-front in cash, no receipt, had to ask twice for the keys.
2. A room in a house in a mountain village (Kazbegi), a couple of days, via booking.com. Pay at the end in cash, no receipt, got keys without asking.
3. An apartment in Tbilisi for a couple of days via AirBNB. Pay in advance by bank card (because of AirBNB), no receipt (maybe can get it through AirBNB?), received the key without having to ask.

I would remark that I use AirBNB very rarely (last time was like 3 years ago), so regarding the last place, probably the payment/receipt is handled according to AirBNB rules. (Booking.com, on the other hand, I use a lot, and until now I always payed at the end, and then got the receipt from the owner of the place.)

So based on this short experience, I would answer my questions as follows, with relation to Georgia:

1. Payment up-front: happens, but not always (it was 50%/50% in my case -- I guess in AirBNB you have to pay in advance by default, so I would not count that).
2. Not giving a receipt: common (at least in the first two places I did not get any, again I'm not sure what is the norm for AirBNB).
3. Not getting a key: weird (in two places I got it without asking, in the first one I had to ask and then got it eventually).