Can the extension of a SETV Single Entry Tourist Visa be done online, or must it be done in person in Thailand?

Does the 60 days single entry tourist visa need to be applied for in your home country?

You don't have to wait for the visa exempt period to finish before extending. You can do it the day after you arrive if.you choose, or at any time when you are close to any immigration office.

Every province has its own immigration office, and it is usually in the major town of that province. There are also offices on the major islands: Phuket, Koh Samui, Koh Chang, Koh Tao, and Koh Phangan. The extension is done in person and costs 1900 baht (cash). The requirements vary at each office, so pop in and pick up the TM7 form (complete in black ink only).

it's for 90 days from most counties check the official website

since Age over 50 Years old and able to meet the Financial Requirements of a Non–O E–Visa based on over 50 Years old ( ฿800k Baht Bank Account Balance or Monthly income of ฿65k Baht per Month ), and, if travel plans are to stay 80+ Days, get the Non–O E–Visa and no need to plan for an Extension of Stay at your local Thai Immigration Office inside Thailand

All extensions are done in-person at an immigration office.

Every province in Thailand has an immigration office. But the vast majority only have a single one. That means depending on where you'll be, it could be hours to get to the office in the province you are staying. Even if there's a closer office, you still have to go to the one that is in the province where you have a TM30 lodged by where you are staying.

Most of the provinces that only have 1 office have that office in the "capitol" city which is just the Muang (City) of the same name of the province.