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Go to the FAQ of Bangkok immigration to find answer to your questions here :

Regarding the fees and forms, you can find informations in the same website here :

And you can download all the forms from the immigrations here :

The Thai Embassy can give the Non-immigrant visa (ED) only three months (90 days), Yes of course you can extend your visa for 1 year at Immigration Bureau in Bangkok, the documents that you need for extension, your university will give you all, and the fee for extension visa for 1,900  Thai Baht
If you need to go out of Thailand to visit your friend, before you go out  you should make your Re-entry  at the Immigrant Bureau,  the fee for Re-entry One-single is 1,000 Thai Baht, Multiple is 3,900 Thai Baht
Important: You must make Re-entry before out of Thailand if you not do that your  Non-immigrant visa (ED) will invalid.
I hope my detail will help you, if you need more information please ask me.
I am also student here now.

bank200s wrote:

The Thai Embassy can give the Non-immigrant visa (ED) only three months (90 days), Yes of course you can extend your visa for 1 year at Immigration Bureau in Bangkok, the documents that you need for extension, your university will give you all, and the fee for extension visa for 1,900  Thai Baht
If you need to go out of Thailand to visit your friend, before you go out  you should make your Re-entry  at the Immigrant Bureau,  the fee for Re-entry One-single is 1,000 Thai Baht, Multiple is 3,900 Thai Baht
Important: You must make Re-entry before out of Thailand if you not do that your  Non-immigrant visa (ED) will invalid.
I hope my detail will help you, if you need more information please ask me.
I am also student here now.

Thank you so much!

One of the 2 problems is solved thanks to your information. 

I am still confused about the extension. My university is based in Belgium and I will only do my internship in Thailand. Just to be clear, I will not be attending a Thai University .

It doesn't need to be extended for 1 year, 1month should suffice.
What documents should my Universty provide for the extension?
I assume my internship needs to provide me with adittional documents as wel.

Documents for your extension:  you need to  request from the company or university in Thailand that you internship with them, they will write the letter to the Immigrant Bureau in Bangkok, give the reason why you need to extend, how long you want to stay. if the police need more document they will tell you when you contact them at that time. Fee for extension 1900 Thai Baht
Any information Call 0-2141-9889  Fax    0-2143-8228  or write to them on Post Comment
or

Thank you for all the information !

First, you would do a visa extension for usually a year.  Your school will, should, provide the paperwork needed.  Then once you have the extension you would need to apply for re entry permit.
Hope this helps