They have killed a lot of school teachers and Buddhist. They may have an agenda beyond hatred for education and a desire of Muslim apartheid, but I have not seen them articulate it.

They do not have a "hatred for education" per se. The Thai education system is highly centralized, and as an ethno-religious separatist insurgency (Malay Muslims), they perceive state education as a Thai Buddhist attempt to destroy their cultural and linguistic identity through their children. Targeting of schoolteachers for this specific reason has been common in many ethnic-based separatist insurgencies, i.e. the PKK's campaign in Turkey (before the capture of Ocalan) stands out in this respect.

I have seen indications that many are based across the border in Malaysia

Funny, this is the second time tonight I'm referring to article in the 17 Feb 07 issue of The Economist:

Thailand and Malaysia: In Need of Help to Douse the Flames
...the treatment of Thai Muslims is a sensitive subject in northern Malaysia, where the country's Islamist opposition is strong. So Mr Badawi will not want to risk providing political ammunition to his opponents.

Co-operation would be a welcome change from the two countries' frequent rows over the Thai south. Mr Thaksin used to accuse Malaysia of sheltering the militants. He was furious when, in 2005, it refused to send back around 130 Thai Muslims who had fled across the border, claiming their lives were in danger.

Since the Thai coup, there have been tensions over a security barrier that the Thai generals want to erect along the border. The Malaysians were miffed at General Surayud's claim that funds were being raised for the insurgency by extorting money from restaurants selling Thai tom yum kung soup in Malaysia....
I have a vague recollection that they may have had ties to Hambali before he was captured which would put them pretty much in the al Qaeda camp.

I highly recommend a read of the May 05 ICG report linked in the first post on the thread (Southern Thailand: Insurgency, Not Jihad), it provides an excellent overview of the conflict.