"I notice side effects like he's jittery, hands shaking, only able to sleep about 5 hours a night so he's often tired, and sometimes it seems he's a little forgetful and clumsy. Not sure if these are classic prednisone side effects or not."

These are very likely to be prednisone side effects. I'm surprised he is able to sleep 5 hours.
I remember being on prednisone doses as high as 100 mg. It told my ophthalmologist that it was great because I didn't have any need to sleep anymore. My not needing to sleep lasted about a week and then I slept for 2 days when my dose was decreased to 60 mg.

Another interesting side effect of high dose prednisone is euphoria --- "a false sense of well being." I learned about euphoria as a side effect from prednisone from my ophthalmologist. I told him that I couldn't see my hand in front of my face and I was pretty sure my left eye was a "lost cause." He shined a bright light into my left eye and reassured me that I wasn't blind because I said the light was white. What I said next was more shocking. I told my ophthalmologist, "if I ended up being blind, I didn't care because I felt great!" That was when my ophthalmologist decreased my dose to 60 mg because I was feeling too good.

Be aware of the Mental Health Side Effects:
"Early in treatment (within several days), prednisone may increase feelings of well-being, anxiety, hypomania or mild euphoria. With long-term therapy, however, people may develop depression. Psychosis, referred to as corticosteroid-induced psychosis, can occur at doses of 20 mg or more per day with long-term use."

I don't want to scare you with all the possible side effects of prednisone. People generally don't experience all of the side effects from prednisone. However, almost everyone will experience some of the side effects. Sleep disturbances, the shakes and euphoria were some that I experienced.
https://www.drugwatch.com/prednisone/side-effects/#:~:text=Early%20in%20treatment%20(within%20several,day%20with%20long%2Dterm%20use.
Prednisone is needed at this stage. However, with some luck Actemra will allow him to decrease his Prednisone dose rather quickly. On a hopeful note, I don't seem to have any serious side effects from Actemra.