1. Here's probably the best resource for sorting out the Russian participles
https://www.alphadictionary.com/rusg.../particip.html

оказавшись = оказаться - to find oneself, to turn out, to prove (to be)
the past adverbial participle translated to English roughly as: "...., having (verb)-ed"

Muscovite, having found themselves in the city on a day off, in my view, carry themselves like tourists.

2. Two things are going on here

A) The T K pairing that you pointed out
так, как - literally translated as: in that way, in which...
The phrase you pointed out uses the "that,which" construction (for when the sentence calls for a noun in both clauses), just with an actual noun on the first end:
тe вещи, что - those things, which

B) The added же + emphatic (и)
This emphatic и is unlike the word 'and', and it shows up often with же (when used as an indicator of similarity)
так - in this way
так же - in the same way
то - that (thing)
то же - that same (thing)
такой - this kind
такой же, как - the same kind, as

У меня есть такой же рюкзак, как и у тебя
I have the same kind of backpack as you do!

Фотографируют те же достопримечательности, что и приезжие.
They photograph those same attractions, that the arrivals do

In some sense the и is used to mark the omission of the verb. Because the theoretical full sentence is "They photograph those same things, that the arrivals photograph."
But most languages don't care much to repeat the verb a second time, but if you just said "..., that the arrivals." then there'd be an awfully confusing pause, so English inserts the "do", and Russian uses this emphatic и.

The reasoning behind why the word for "what" is so perfect to use as the word for "..., that ..." is a little complex. Regardless, most European languages use that word, and Caribbean (Pirates of the Caribbean movies too) English uses it. You can here them say "that, what".
The basic answer is that if the что is in the main clause, it is "what?", and there will be a question mark, and if it is in a subordinate clause (which you can think of as a sidetrack) then it is "...,that...". The beauty is that those two categories are mutually exclusive.