"I see him clearly."
Translation:Jag ser honom tydligt.
tydligtvis is always an adverb, but tydligt can be either an adjective in the ett form, or an adverb. It is a much more common adverb than tydligtvis, and there's a difference in meaning – tydligt means 'clearly' as in 'in a clear fashion', but tydligtvis means 'obviously', so it wouldn't describe 'how you see him' in this sentence, it would be a comment on the whole phrase instead.
If you'd like to say I see him clear, meaning 'I see him and he is clear' that would be Jag ser honom klar, because then it would be an adjective describing him, not 'the way in which you see him'.
I'm not quite sure either, but I think the adverbial normally goes last in sentences, unless it's a clausal adverbial, which this isn't. I try to use this http://www.student.umu.se/under-studietiden/studieverkstad/skriftliga-uppgifter/skrivrad-och-sprakhjalp/ordfoljd-i-huvudsatser-och-bisatser/# but there are so many cases that aren't covered.
I was thinking more along the lines of "obviously" (see Arnauti's comment above). You know "The Princess Bride"? "Australia, as you know, is populated by criminals, and criminals are not used to being trusted, as I am not trusted by you, so I clearly cannot choose the wine in front of me." Or something like that ...
I agree. I think I rarely ever use the word tydligtvis at all myself. Tbh I think it's one of those words that I don't really know what they're good for. There's tydligen, and there are the ones you mentioned, but tydligtvis?? I think it sounds somewhat formal without sounding elegant or intelligent. It's good for being snotty I guess… (Seriously, it's hard to think of all possible contexts, so there are probably some where it makes sense to use this word. But I bet they're very few).