When I did ThaiPass in January it had taken 3 weeks. Things have changed and supposedly its more streamlined now but I would not wait until last minute.
I prefer not use an agent but sometimes it is worth while. I think also depends a little on the immigration office you intend to use. I find Pattay's immigration office to be challenging while Bankgok and Phuket a little more user friendly.
If you have the time and patients, do it yourself. If you have special needs, value your time, or rather not be bothered, use an agent.
If you choose to use an agent, get a recommendation from some who has used that agent for the same purpose you require.
I think overstay wont necessarily be an issue if your partner never wishes to return to Thailand. It probably won't cause them too much trouble in the future. However, it could cause them some trouble and I would recommend avoiding the trouble. Cut the trip short by 4 days, 3 days if they show-up at the airport the day of, or 2:00am next day. Or find a legit visa option.
I doubt you will have troubles with doing visa exempt if you show history of leaving (not extending). I am in the process of doing a marriage visa and its becoming a real pain.
I use AirBnb for long term rentals. bahtsold.com may also have options for you. If you are not familiar with the island I would recommend coming to see the place before committing to a rental for months. PS The west side of the island seems more popular with foreigners