Could Jacob Become a Vampire?

He def wouldn't become a vampire. He would most probably die depending on how severe it was. Jacob said in Eclipse that if a vampire bit him, or one of the other shape-shifter, it wouldn't kill him, but if he already had a cut of something to that affect(combined with the venom) that could be fatal.

Can Jacob's dad turn into a wolf?

As a result, Billy grew up hoping a vampire would cross Quileute land so he would phase and become a wolf like his grandfather, but that wasn't the case. Billy and Jacob's ancestors would transform when they reached manhood, but as years passed, this changed completely.

Does Jacob age with Renesmee?

When they phase into a wolf for the first time, they stop aging. There are plenty-a-plot-holes here BUT if Jacob maintains a regularly scheduled shape shift, he can essentially stay young forever. This is a good thing because Renesmee will be Jacob's age physically in a mere few years.