Edit: What about buying a local dirt bike in Malaysia. Could I take that one around Laos Thailand and Cambodia?

Well - SE-Asia is a bit complicated and it seems to constantly change.

Its possible to buy a bike in your own name in Thailand - although it might take some time and patience. Just as you know. It could be more conveniant to hire one....

Then to your questions....

1. If the bike is in your own name or if you have a verified document from the owner a crossing from Thailand to Laos shouldnt be a problem. There are however a couple of things that must be known. Xayabouli province in Laos seems to have stopped any border crossings for motorbikes and bicycles - so you cannot get into Laos using that province - but there are many other places to cross over so that shouldnt be an issue.
Thailand have banned small Lao registred motorbikes to enter Thailand - thus Laos has done the same with thai plated bikes. Not 100 % official but and you might get through but I wouldnt count on it. A 250 ccm is considered a big bike in that part of the world so you might have to go for a 250 at least.

The only border crossing between Laos and Cambodia seems to have been closed for foreign (included thai) plated vehicles in the direction from Laos to Thailand. Permits is needed its reported. In the opposite direction from Cambodia to Laos there seem to be no problem.

From Thailand to Cambodia it seems that there are two bordercrossings that will let one through without any hazzle - namely the O'Smach bc in Thailands Surin province and the Hat Lek bc in Thailands Trat province. The other provinces seems to from reports tondemand permits. However - this is not something thats 100 % - so be aare of that.

2. Yes its possible to buy a bike as a foreigner in Thailand. But it can take some time and certainly some patience. And if you buy a 125 cc bike you will probably have big problems getting it into Laos. Permits are not needed going from Thailand to Laos on a thai plated bike. Going from Thailand to Cambodia permits seems to be needed at most bordercrossings except the 2 I mentioned above.
To sum up - if you plan your trip well and have a bit of luck you will not need any permits to travel Thailand, Cambodia and Laos. For example Start in Thailand and ride into Cambodia first and use one of the two bordercrossings mentioned above. Ride a bit in Cambo then cross over from Cambo to Laos. Then travel Laos as much as you like and then cross over to Thailand again.

Maybe our thai living member Recom273 can add more?

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In the end everything will be fine. If its not fine its not the end....