Learn Icelandic: Uppistandarinn talar allvel. - The stand-up comedian speaks fairly well.

It is a false friend! While it looks like the English word "all," the Icelandic prefix all- actually means "fairly," "quite," or "rather." So allvel means "fairly well," not "completely well." If you wanted to say "completely," you would use a word like alveg.

Can I attach all- to any adverb, and does it change how the word behaves grammatically?

You can attach all- to most common adjectives and adverbs of degree or quality (like allgóður for "fairly good" or alloft for "quite often"). Because vel is an adverb, it naturally does not decline to match a noun. Adding the prefix does not change this; allvel functions exactly like vel in the sentence.

Where does allvel sit on the scale of praise compared to ágætlega (fine) and mjög vel (very well)?

Allvel sits comfortably in the middle. It is slightly better than ágætlega but not as enthusiastic as mjög vel. It translates best to "quite well" or "fairly well," often implying that the speaker is giving a solid nod of approval or is perhaps pleasantly surprised.

Uppistandarinn is clearly derived from the English "stand-up." How is this word constructed in Icelandic?