Ddd l5, djd right tmj, chronic sciatica and back pain, hypermobile fingers, hashimoto's thyroid. Could there be an underlying disease connecting these?

Look at Ehrler- Danlos syndrome. There is a wide spectrum of manifestations for this genetic disorder. I don't think hashimoto's is related though.
Yes its possible. Seems like you need to speak with a rheumatologist and a musculoskeletal spine specialist that can help create a specific treatment plan for you.
Rheumatoid arthritis. May be worth it to see a rheumatologist to see if you have rheumatoid arthritis or another autoimmune condition...