There is no such word as "fastily." The adverb "fastly" means securely (made fast or attached). The word fast is the adverb used to mean speedy or quick.
Adverb
Yes, thankfully we can use it as an adverb. It is the adverb form of the adjective thankful.
You would have to use one of the adverb forms "thoughtfully" or "thoughtlessly."
(You would have to use the adverb, which is tenderly.) She touched his face tenderly.
Formulaically is the adverb form of formula.
As an adverb .
no
Like this: I did my history homework yesterday. That's how you use yesterday as an adverb in a sentence
The adverb would be "inalienably." But it would have very limited use.
Adverb: I had a hat on. Preposition: I didn't have any money on me.
In the English language, "fire" has no adverb form.
As your question indicates, "entirely" is an adverb; it doesn't need an additional suffix.