You are good. As long as you have a flight booked to leave Thailand within the initial length of entry (45 days currently for visa exempt) before you board your flight to Thailand.
Also note that you are technically required to have 20,000 baht (or other currency equivalent) in cash in pocket. Immigration rarely ask, but it IS a requirement.
You can go for it now. Some places have different rules etc, but take your 1900 baht, take your form, passport and go now. Plus, it gives you time in case you forget something etc.
The 6 months starts at some point during application. Then you have multiple 60 day stamps (leave and come back) within that time frame. So, if you apply on March 30. Get it April 5th, you have until October 4th to enter. If you enter April 5th, you have to leave (or extend) by June 4th. Go somewhere, come back. Repeat.
Thailand is... different. But since it is a European owner, let him know that you will fix the heater, and take it off next month' rent. He may agree. Otherwise, you may just have to get it fixed on your dime. It probably won't cost you too much anyway.
And remember. This is Thailand. Getting mad at the locals will get you nowhere.