If it is a metal carport some companies offer leg extensions. Of course they do need to be installed properly for stability and securing the roof so the wind does not pick it up. I would say best done by a carport installer with a couple of guys rather than DIY but with some research it might be attempted as a DIY. If it is a wood carport it is also possible to extend the height by building a short section of wall just under the roof. But that means releasing the current fasteners that tie the roof to the support post, then jacking the roof up on temporary supports, adding the new wall section in, fastening it to the post and lowering the roof onto it and then connecting the roof to the wall section. Quite a bit of work and some expense involved but raising the height can be done.
Depending on the damage to the ceiling and the type of structure that holds up the ceiling. You might be able to install some 1/4 thick sheet rock right over the current ceiling material without removing the old ceiling. You need to go to a sheet rock supply company to get 1/4" thick sheet rock. You don't have to be a contractor to buy from them. It is much easier to handle that weight of material than the 1/2" thick panels.
You might not have had the experience for these things but I have, I have remodeled a number of houses including skimming a ceiling with 1/4" thick sheet rock over a damaged ceiling. A bit tough on your neck and shoulders to work overhead for a lot of hours that are needed to get all screws put in and all the taping and plaster mud as well as sanding, priming and painting done. At 69 I can't do it anymore without first taking half a mild muscle relaxer pill and some OTC pain killers before I start working. Too much wear and tear arthritis in my neck, shoulders, hands, etc from years of doing those kinds of projects. It would likely take me 2 weeks of 24 hours of intermittent labor to accomplish what a young guy can do in about 12 hours of so of intermittent labor time over 3 or 4 days. I can do it, but not without a fair amount of pain.