Can I leave flouride gel in my clear retainer and sleep with it overnight, every night, or will that damage my teeth?

Yes. It should help prevent issues with your teeth and will not damage them.
Yes you can. Topical Fluoride gel in your retainer will not damage your teeth, but it will help strengthen the enamel. Fluoride gel, however, is not meant to be swallowed. Place just a small amount in you retainer so that any extra does not squish out when you put the retainer in.
Follow instructions. A clear retainer (like one used for bleaching) is great for delivering Fluoride gel. Fluoride gel is also a great adjunct to regular daily hygiene. I would advise you to closely follow the instructions of your prescribing doctor. The risk will depend on the type of gel, but you shouldn't create your own regimen with high concentration fluoride. Avoid injesting it, just have it contact teeth.
Fluoride gel. Some Fluoride gels are more acidic and should not be used long term. Check with your dentist to make sure you are using a neutral Fluoride gel.