Officially you need an onward flight for the end of your visa. It's the airline that may check this although not always so you're taking a chance of having a problem at check in.
It is worth applying for tourist visa in your home country. Most Thai embassies have made it pretty straightforward and painless. No hassle when you arrive
It's the airline at check in who will ask about onward flight. As already stated if you have correct visa in place it's not a problem but bare in mind a tourist visa is issued on the understanding you will be leaving the country again so onward flight would be expected.