One way flight?

Belize requires an onward ticket and sufficient funds for your stay. Belize Immigration, on your arrival, doesn't check for that very often. It is actually the responsibility of the airline to ensure all passengers are complying with entry laws of the flights destination. Some airlines are more strict than others. I do know of people that were denied boarding.
Once your 30 days, or whatever date you are given upon entry, are up, immigration can ask for proof of onward/return ticket and proof of funds.

I would say the short answer is yes you can, we did. I'm not sure about actual legalities but when we moved from Canada just over a year ago we even brought our pet (with proper paperwork and permits), we were never asked for a return ticket. On arrival at Phillip Goldson we were asked our length of stay, we told the immigration people that it was our intention to stay and they issued a 30 day visitor visa. That is the maximum length of visitor visa issued. Every month as required we renew our visitor visa at the immigration office and will continue to have to do that until residency that we have applied for is approved.

Sure can. Try cheapo air.

I do know that some people have said they are flying in and doing land travel through Mexico or onto Guatemala who bought one way.

yes, I do on any extended stay