I know of people that just sit and relax in certain 5 star hotels and watch the world pass by, dress nicely and enjoy the scenery in the Oriental Hotel, if you lucky you can watch a Indian or Chinese Marriage.
there are plenty of things to do, one of my favourites is still the riverboat which stops at many places. You can also explore certain areas like Sukhumvit, Khao San, Grand Palace, Pratunam, Siam. Bangkok has also a nice collection of Traditional Thai houses , plenty of temples and markets. But do not overdue it, Bangkok is hot, so take plenty of water with you.
in principle he can pay overstay at airport, but depending on his nationality and destination country there might be a problem. No every airline or country wish to let people board or enter while having an overstay.
not every Thai Embassy in Europe will issue you a visa, a few have restrictions which I do not know in details. Best to select an area and send request to that Embassy or (preferable) Consulate.