If you use rentalcars.com site and put in your criteria, it will list the companies. I have done a one way rental with Thai Car Rent before, no extra charge.
Yes one way available at some companies. Use rentalcars.com to do a search, putting in the different return location. The longer you rent, the better your chances.
It is normal here for family members to camp in the hospital. If the mum has no other family support I suggest provide a small roll up mattress, spare clothes or do laundry, meals, and moral support. Hope there is a good outcome for all.
Go a different visa route. Come as tourist get 30 day visa exempt entry. Apply for 90 day O Visa in country. Extend that by 12 months in country. Over 50 and have finance.