Yes, you can get a reentry permit on a tourist visa. 1000 baht for single or 3800 for multiple. You must buy the permit before exiting the kingdom.

Airlines can make their own policies and if the airline's policy is that you need proof of exit travel to get on the flight, then you will have to proof of exit travel or you are not getting on the flight. Since it is airline policy, management can of course waive that, but if they won't there isn't anything you can do except buy a ticket on the spot.

They have your name, nationality, birthdate, and photograph in their database. They can cross-reference that.

Proof of funds, proof of accommodation, proof of exit travel.

The immigration officers can see every entry you have ever made in their database. Have your three proof-ofs and a plan B.

If you want a tourist visa, select tourism. If you want an education visa, select learning.

Multiple Entry Tourist Visa valid 6 months from date of issue. Doesn't require any investment in Thailand, however requires $7000 USD bank balance in your home bank. You can enter and exit the kingdom as many times as you want during the 6-month validity period and stay up to 60 (90 w/extension) days at a time.

You are applying for the visa, not your wife. The money must be in an account under your name.

Note that they count your arrival day as "day 1," so make sure you are counting your days correctly. Agree that you should not cut it too close--what if your flight is cancelled or you don't make it for some reason? As you will already have an extension your options for contingency are pretty limited.

No. The immigration officer, at their discretion, may not charge you a fine for one day overstay but your passport will still be marked as overstay.