Here's my summary of your issues, and some potential ways to lessen them. Just IMHO..
1) Person to respond to inquiries & questions when you are vacationing around the world ;)
I think the first thing is eliminating all the questions you can.
a) Simple things that are easy to operate, as you said, would be great.
b) Pre-address existing questions through email & FAQ book on-site.
Go back through your email once and look at all the questions. Copy and paste the answers, then put any of 2 of the same questions towards the top of the list. Make a PDF and save in a folder, then print and put in a laminated binder clip thingee in a nice leather binder (OfficeDepot or Staples). I even got nice paper to print on. This could be for the operation of appliances & lights.
c) Use automated email.
Save an email w/ the PDF you made for the unit, and email it to anyone with your other PDF (airport instructions, entering the unit, how to use the keypad, stuff to do, etc.) when they book the unit. Also, set a delayed email reminding them about checking out the day before they are supposed to leave.
d) Use automated email #2
You can set up a different email address for each unit, and have an auto-reply with all the FAQ's that automatically goes to them when they email a question.. "While we strive to answer all questions as quickly as possible, we spent significant time compiling these FAQ's to assist guests' needs as quickly as possible. Use the search function (or "Ctrl+F" on the keyboard) to search for your issue. For example, "Dishwasher". Or "towels".
e) Pay for what it's worth. It has to be worth your property manager's time and effort to "babysit" your guests, if that's what's necessary. If the margins are there, it's a cost of business and enjoying the world. Hard to have ALL the cake and eat it too..
2) Reliable cleaner who can identify rule-breaking
a) You need to pay for what you want. If you want them to drive out and show up for a brief period, and have it perfect, you need to pay for it. I asked a neighbor and found a local cleaner. She's done some AirB&B. I got a quote from her. Then I upped her pay, and told her I'm happy to pay for a great job, but it was extremely important, etc. She was happy. So far, so good. Charge it to the guests in the cleaning fee.
3) Reliable handyman who can address issues in a responsive time frame
a) Again, you pay for what you get. I made some financial sacrifices and committed to giving him a certain amount of hours each year basically - and it's worth every penny. Good people are worth it. If the margin is there, it's just a cost of doing business.
4) Tenants don't always realize what day they're supposed to check out
a) See 1.c. above. My business email has a feature where I can send a delayed email. This should pre-address this issue. Just book the maid, send the welcome/FAQ email, and set the scheduled email when you get the accepted booking.
5) Don't know which unit it is
a) See 1.d. above. With an email for each unit, all forwarded to your phone and/or outlook in one mailbox, you can see which inquiry is for each unit, and again, have an automated FAQ response for any emailed inquiry.