Yes, you can use it as an adverb, just not like that ("fuckingly" is not something you would ever hear). It is often used to intensify or amplify the meaning of a word. You might hear something like "I was totally fucking shocked that I won a fuck ton of money in the lottery" Or simply "I don't fucking care". But of course, it is a bad word and you shouldn't say it blah blah blah...haha.

Not in polite conversation, no. (Although I daresay your audience may overlook that if you are sharing the bounty ...) As with many other intensifiers ("bloody", "damned", etc.), the adjectival form works as an adverb, too, so "fucking" may be the word you had in mind.

Nope, we don't say that. People do use just f***ing, without the -ly ending, though!

I am an American adult. My son is 30 years old and I don't allow him to speak that word in my presence. If I were to hear you use "f--k" in any form during a conversation with me, I would immediately ask you to not say that word in my presence. It is vulgar and impolite and is used mostly by people of limited vocabulary who don't respect authority. Please do NOT try to emulate Americans by using this word. Adults will not like you, though you will fit in with immature, irresponsible teenagers just fine. That's my opinion.

Yes, your favourite English swear word can be an adverb, but not with an 'ly'. If you just remove the 'ly', your sentence is perfectly correct. Taboo words are wonderfully flexible.