Is it possible to drive from the US through Mexico to Central America? What route is recommended, and how long would it take? What safety or immigration/customs concerns would you face?
Depends on which Central America country you're visiting, but for the most part the Pan American highway is your safest bet. Also, you probably want to read the answers to this question: Driving from San Antonio to Copan Ruinas, Honduras
You can certainly drive it. You can get as far as Yaviza in Panama, but no further.
Mexico is very picky about the paperwork needed for your car. I'm sure the other countries vary.
Expect to have to pay some bribes sometimes, for this I was recommended to take lots of cigarettes to be used as "gifts" before officials start to give you grief. Don't drive at night after Mexico, and you probably don't want to do it in Mexico either.