Does this verb mean both to support physically (like support someone with crutches) and to support someone financially (support with money)?
That's true, but notice that if you want to say that someone supports someone financially as in pays all of their living costs, the word for that is försörjer.
It's crazy how many words you already basically know if you speak German and English while learning Swedish like stöttar = stützen, väljer = wählen, vill = vollen...
Indeed. Due to the very heavy influence of the Hanseatic League in the middle ages, Swedish is basically the bastard child of Old Norse and German.
I think the vikings could be described as being "Germanic" and stuff if "Crusader Kings 2" taught me anything...
What verb is more prevalent? I have a book of swedish verbs and tense conjugations, and the verb "to support" is shown as "stöda" and "stödja". Is the use of "stötta" more common in speech than the other two?
stötta is a somewhat colloquial. For more abstract meanings, like if you'd say things like 'this supports her argument', or 'I support this suggestion', we only use stödja or stöda (those two are just spelling variants).
stötta is more common for emotional support
These verbs are used in many expressions and which verb is preferred can vary a bit depending on the specific expression.
Would stöttar also be used in reference to supporting a sports team (for example)?
It could be, but not in the sense of rooting/cheering for a certain team, in which case you "hejar/håller på" a team (not a phrasal verb, the particle isn't stressed!)
why the voice pause in the middle of the stöttat in the sentence? han stö-ttar mig
Hm... English does have this feature. Perhaps it is not as consistent as Swedish though. I remember, for example, when learning the word "bookkeeper" my teacher stressing that there were two K's, like "book-keeper," not "book-eeper."
Is stöttar also used in supporting a building or mine so it doesn't collapse?