So if you say the shoresh twice it makes the word an adverb, as in לאט לאט and in סוף סוף?
It's not a shoresh. Think about it like the use of adverbs in English: "He drinks the soup slowly slowly".
They're slightly different to one another.
Sof סוף is a noun meaning "end", and doubling it up makes it an adverb, "finally". To emphasise it, you can say סוף כל סוף, "at long last", "finally".
Le'at לאט is already an adverb, but doubling it up is quite common, and probably more common than just saying it once. I can't say why this is.
No, it's just some sort of emphasis on the slow speed, or the fact it's step by step.
Yes we feel it the same way in Czech: a soup is eaten not drunken. If a child drinks soup directy from a plate so it is considered to be indecent behavior.
Not really, that would be מעט מעט
Or צעד צעד (step by step), which for some reason also goes as פרה פרה.
No. Maybe "באופן איטי" should be accepted (using the adjective - not the adverb). But לאט is much more common, shorter and generally preferable :-).