I've been working on a 71 chevelle for a year and now's time for the interior. I only have 3 of the support rods and the support plates on the edges are in rough shape and I have no idea how to put it back together. How much does it cost for a professional to install the headliner? Does a professional still need these pieces from me, or are these common parts for a pro?

You can buy new headliner bows, and they are specific to a chevelle so you would most likely need to find replacements, or buy a whole new set of them since each one is different.

what supports the sides, front, and back?

typically the sides and front are held by chrome trim, and the back should fit around the sail panels and tuck under the rear window trim.

On a '71 it is plastic trim.

Yep !!!

I just finished installing a headliner in my 68 and it can be D.I.Y. without much worry. If you can get the bows and get information on where each one goes it is fairly easy. Some things I learned:
1. My headliner came with about 3-4" extra material on each side. So you will have to carefully cut back the "pocket" the bow slide through just enough to get the ends in the frame.
2. Carefully count your bows. I thought I had a 5 bow and ordered the same when in fact I had a 6 bow. I actually left one bow out and it looks fine.
3. Wherever you get your headliner from-Find out which end is the front!! I hade mine all set in to find out it was backwards. DOH!
4. Do a "search" for headliners on this web site and you will learn a lot.
5. Go to an office supply store and buy about 50 med. size paper "clamps" (about $2.50 per 25) and use them to fine turne your liner before glue... A life saver.
6. Take on the challenge and do it yourself!!!! Pride baby.