Every case is different, and this is a decision you should be making with your doctor. I am assuming that the compression fractures are old injuries and healed to whatever extent possible, so you better understand than anyone what hurts, what doesn't and how much you can handle.
Having said that, I have 5 broken bones (not compression fractures - complete separations) in my back as well as arthritis and continue to "ride". I am still determining what my limitations are, as these can change on a daily or even hourly basis.
From experience, I can tell you that strengthening your core is about the best possible thing you can do for yourself, for riding as well as every day living. If there are no pilates classes near you, try some you tube pilates videos for lower back pain. They strengthen the muscles in the area and provide a sort of internal massage action. Highly recommendable.
Once you have an OK from your doctor (or not, because that would mean another CT scan and I never got one), discuss your options with a therapeutic riding program. Perhaps you are mostly interested in interacting with the horse, so a good long brushing, braiding etc. would be fun for you. This could be followed by just trying to mount, or sit in the saddle and caress/interact with the horse. There is nothing wrong with that. If you are able to work at a walk, fabulous! Arm movements could be added. Realistic goals should be initially made, however. Five minutes, ten max at the beginning, followed by lots of lovin' and treats for the horse. Additional 5 or 10 minute increments could be added in subsequent classes. I recommend these baby steps for you because they are what worked for me.
As far as driving goes, that also gets complicated. I would not physically be able to withstand much braking, but it is a valid option until you experiment with it yourself. Here, the size of the horse and cart would be key.
I wish you much luck and especially patience (which is something I am still working on myself!).