Re: Low T3?
You haven't had the best thyroid tests. The TSH and free T4 are good, but you need free T3 done as well. The test you had is total T3, not as accurate as free T3. Ditto for total T4.
The other tests you had are not accurate at all and are basically worthless. Your (total) T4 is definitely not "high". It's probably too low, given the 20% inaccuracy rate it carries. Its level reflects itself in your low-normal free T4. Yes, your free T4 is too low... as is your TT3; that means you do need the Cytomel, but you likely need a bump up in your Synthroid as well. (You don't say how much Cytomel you were prescribed. That would make a difference in where the Synthroid dose needs to be.)
Your liver and other tissues apparently don't convert T4 to T3 very well, and that's why the T3 level is low. About the only thing I'm aware of that might help that conversion is selenium, at least 100 mcgs/day. But when that fails to help, T3 supplementation - as you've been prescribed - is a must.
Your depression might indeed be caused by low T3. Speaking only for myself, I wouldn't even consider taking an antidepressant until my thyroid levels were optimal. The AD can create its own huge set of problems. Why even go there if it's not needed in the first place?