Regarding the script. They have the same basis from India but it is misleading to Westerners because they will think that it is similar in the same way that Russian/Portugese/German/English all have the same script.
While they Burma/Thai/Cambodian all share the same source for the script, knowing one script doesn't help in being able to make any sense of the other scripts, and even if you could pronounce the words, you wouldn't understand them because there is no relationship between the vocabularies. I can read Thai but cannot make out any letters in Burmese or Cambodian even though they all borrow from Pali as you note.
Thai and Laotian are related and part of the same language group, the Thai language has 5 dialects and the Laotian dialect is one of them. While it is possible for Thai and Laotian speakers to understand each other in the same way that Portugese and Spanish speakers can understand each other it is impossible to read each other's script, unless you have learned it.
Now the exception to this is that when it comes to religion and sacred language the vocabularies of the different languages are much more closely related than the common language, having all been based on Pali.
Migration is a fascinating question. It appears that Thais migrated from China. There is a large Thai speaking minority in China and they speak a dialect that is remarkably close to the central Thai dialect (and easier to understand than the other major dialects) even though they have been cut off from the larger Thai group for centuries.