As quickly as possible can be translate as
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 :) )