1. I want something to write.
2. I want something that I can write.
3. I want something to write on.
4. I want something that I can write on.
5. I want something to write with.
6. I want something that I can write with.
(Is each pair the same? Can we use other auxiliaries instead of 'can'?)
No, they are not all the same. You want something to write? That could mean you need a topic or an idea that you could write about. You need something to write on? That could be a topic, a piece of paper, a slate, a chalk board, a white board...it's very vague. You want something to write with? That could be a pencil, a pen, a marker, or any writing instrument.
As to the use of "can", yes, they all mean the same. "...that I can write with..." or "to write with", both mean the same. You could say "with which to write" or "upon which to write", but "that I can write with" is simpler and to the point.