As quickly as possible can be translate as
möglichst schnell
or
schnellstmöglich
Is there any difference between the two versions? What other adjectives have the second version (i.e., _________stmöglich)?
As quickly as possible can be translate as
möglichst schnell
or
schnellstmöglich
Is there any difference between the two versions? What other adjectives have the second version (i.e., _________stmöglich)?
möglichst schnell has a more friendly touch: Please make it as fast if possible. You could use this, if you need it fast, but you have no authority to force it.
schnellstmöglich expresses more urgency: Make it as fast as possible! This could be used by your boss if he needs it urgent.
For your second question:
I selected a list of ~stmöglich from a text database and get this result (in bold my subjective selection of common words):
This kind of phrasing arises from the superlative form of an adjective.
infinitive, comparative, superlative
good, better, best e.g.
are the three states an adjective can have.
In this case:
schnell, schneller, am schnellsten
By swapping these to phrasings, as you did, the superlative state just gets passed/swapped from one word to the other
schnellstmöglich or möglichst schnell
Another phrase which might be heard is: als bald möglich or möglichst bald (I think my father uses this from time to time :) )
schnellstmöglich means "as fast as possible"
while
möglichst schnell means "fast, if possible"
therefore, as knut already said, "möglichst schnell" is not as urgent as "schnellstmöglich" and more polite in that context.