What is the difference between 毕竟, 究竟, 到底, 终究, and 总归?
The only real answer to this is reading a stack of Chinese texts and picking it up by osmosis. It's a gut-feel thing. But there are at least two cheats you can use to build intuition fast. Check out the raw frequency on google ngram viewer (to look at all your terms at once, make sure you're typing English commas as separators not Chinese ones). Two interesting things there: 终究 and 终归 are dramatically rarer than the other three, so they're probably formal or jargon-y. Worth avoiding unless you have extra special motivation. Also 到底 spikes massively from the 60's to the 80's. Sometimes google ngrams spike for no reason at all, or butterflies-flapping-their-wings-in-brazil reasons, but also, things that spike like that over that time period are often anti-pretentious earthy language of the 老百姓. （See also 革命，前进， and the spectacular crash in the frequency of 之... these can't all be brazil butterflies, right?)
A better way is to check out usage-in-context, ichacha.net has a good collection of these, there are plenty of other good sources. From their example lists you can see things like the way 究竟 appears very often in rhetorical questions, where 毕竟 is much more associated with statements.
"You're my mistress, not my wife" feels very different from "However did you get here without a car?", although they're both sort of in the "after all" zone, the first is much more typical of 毕竟 and the second is much more typical of 究竟. Check out the full lists on ichacha and see for yourself!