Learn English – Graduate student in or of

As a Painter, I find that it is a good thing to have a standard signature that sets the piece apart as distinctly yours. I take this cue from Egon Schiele; check out his paintings and look at how he signs/endorses his works. As to writing a dedication, writing it on the work itself seems like it would take away from the piece itself. Giving a gift of art, and ensuring its value seems to be important to professionalism.

That said, if you properly mount or mat the piece, I would think that the surrounding border would be fair game to write whatever you want. An elegant signature on the piece, and a "To {la la}, {yada yada message}" on the framing sounds like a professional solution that reinforces the personal meaning in gifting the art.

Well, if you're not a student anymore, you probably shouldn't be referring to yourself as a student at all, and just say that you "graduated with a bachelor's degree in ...". If you're an engineer or architect, you could say something more along the lines of "I'm an engineer / architect, and I received my undergraduate degree from ...".

However, the word to use would be "graduated", because (as you appear to have correctly surmised) a "graduate student" is a person in graduate school (so, a master's or doctoral student).