There is no big problem crossing the border by bike, just the usual border formalities (visa, etc).

Myself, I would come up the west coast, primarily because there have been some episodes in the eastern province by separatists (though tourists haven't been targeted). Plus you will find more options for tourists along the west coast (hotels, guesthouses, food, etc).

Actually drivers are far more bike friendly in Thailand than North America, Australia/NZ or even Europe. They are used to sharing the road with slow vehicles and always give cyclists the room they need.

To the uninitiated the roads seem chaotic, but there is a pattern and rhythm to traffic in Asia. If you adopt to that pattern, then cycling (and even driving) is quite simple and safe here.

Coming up from Malaysia, you are limited to your road choices, as a narrow isthmus with a mountainous center is your terrain, limiting north south roads pretty much to the coast. There are stretches where you can get off the main road on parallel side roads. But part of the time will be spent on the main highway.