If you rent a bike you have to leave your passport with the rental company so you will most likely not be able to cross borders.
If you buy in Thailand its complicated with all the paperwork. Sounds like a PITA to do unless you know a local.
Malaysia is a popular place to buy and you have no worries going anywhere pretty much. Vietnam is an option to buy as well (just know the paperwork will not be in your name) There is a company in Ho Chi Minh that sell reconditioned bikes and they have a buy back policy after your trip.
As for bikes as long as your not in a rush or want to do serious off road stuff then IMHO its a no-brainer. Buy a Honda wave. Can go just about anywhere, its cheap even new and absolutely any mechanic can work on it all through SE asia. Its a little workhorse.
I recently did 2 weeks in Laos on an XR 250 and had quite a bit of trouble with it. Mechanics there pretty much work only on scooters. They were no help whatsoever. Even in the remotest of places you can find somebody to fix a wave.
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