1. You should have re-keyed the property when you purchased it. Who knows how many people have keys to that door and yes it seems completely unreasonable for you and the tenant to not have a key to enter through the front door.
2. While you're changing locks you should do this door as well. Go to home depot and buy a 2 pack of keyed alike locks in a set for $50 and do both doors.
3. Tenants are responsible for their own window coverings. If they wanted blinds they should have raised that concern prior to signing a lease.
On to the rent issue. I wouldn't cut them any slack because whatever you give them they will try to take more. Post a pay or quit notice when they're late and if they don't pay by the expiration date let them know that you will be evicting them. I don't do cash for keys because I'd rather give my money to the attorney than some deadbeat, but that's my own personal opinion. An eviction costs me $800 all in for an attorney to do it for me and takes about 3-4 weeks here. It's not a huge deal and once I sign the papers and give the attorney a check it always feels like a load off because I know I'm getting rid of a headache.