Okay I did a lot of research online and it seems they are synonyms meaning "here" (in place or time) with the following subtle differences:

тут It is more informal/colloquial. You don't use it with your friend's grandparents for example. It is also used to mark changes at a point in time. "Тут заходит дочь." And then the daughter came in.

здесь More widely accepted formal situations, but doesn't sound formal (has a neutral feeling).

[ Shady_arc excerpt ] Тут is used in set expressions where you cannot use здесь:

тут же = right away, immediately
И тут... = And then ... (narrative device, so to say)
не тут-то было! = far from it! (in narative, when an expectation is broken)
тут и там = here and there (arguably)
forum.duolingo.com/comment/11441461/Difference-between-тут-and-здесь

[ тут ] [ здесь ] are frequently synonymous but in many context they are not interchangeable.

Ни тут ни там ‧ neither here nor there

Here ‧ тут здесь тута вон вот сюда отсюда

где? ‧ тут здесь (location - not motion)
куда? ‧ to where ( motion )
откуда? ‧ from where ( motion )

where there here
whence thence hence
whither thither hither

тут / здесь ‧ is, are, was, will be: here ( no motion )
сюда ‧ hither ‧ to here ( motion to here from elsewhere )
отсюда ‧ hence ‧ from here ( motion to elsewhere )
context.reverso.net/translation/english-russian/here

там ‧ is, are, was, will be: there ( no motion to nor from )
туда ‧ to there ( motion from elsewhere ) [ там ‧ there ]
оттуда ‧ thence ‧ from there ( motion to elsewhere ) [ там ‧ there ]
context.reverso.net/translation/english-russian/there

Yes, it's right.

‧ здесь ‧ Да, мо́жно, но, по-ру́сски наре́чия обы́чно ‧ In Russian, adverbs are usually placed before the verb they modify. However, in many cases the position of an adverb does not impact the meaning ... ‧ masterrussian.com/adverbs/introduction.htm ‧

The default position for most adverbs is before the verb or adjective they modify, especially if they are short, single adverbs. ‧ ‧ However, adverbs may turn up anywhere in the sentence, but initial position is reserved for emphasized adverbs. ‧ ‧ www.alphadictionary.com/rusgrammar/wordorder.html ‧

живу́т ‧ Жить conjugation ‧ cooljugator.com/ru/жить ‧

Конечно)

http://cooljugator.com/ru/%D0%B6%D0%B8%D1%82%D1%8C this website seems super useful (I was confused on this one too).

живу would be "I live," живут is "they live". i posted a link to a site with full conjugations above

ь is not pronounced but affects pronunciation, whereas ы is a "letter" unto itself. As with other words that use one or the other, in this case you'd just have to remember how здесь is spelled. It might be helpful to remember that it has only one syllable, but ы would add another vowel sound, making it pronounced "zdess-ih" (which is not a word).