A Y test, such as at FTDNA, for Y STRs (short tandem repeats) will get you matches *only* for the Y chromosome.

If that is what you want, then testing your brother (if he is your full sibling) is just as good as testing your father. Yes, there could have been a mutation leading to your brother, but that should not be significant.

Matching Y chromosomes may or may not yield you many results. It depends upon all the other males who have tested.

Surnames are often attached to the Y chromosome, in our culture, but for most people that is a rather recent development.