.
1) The next day, they find them fallen on Mount ***.-- I think that most grammarians would classify this has a nonfinite clause acting as a sentence adverbial.

2) ... came to a man who had been too exhausted to follow him. -- In this case, I think the use of the word "too" gives a clue as to what it is, an adjecitive - agree?-- Yes and no.  A case could be made for passive voice:  'He was too exhausted by the climb....'

If you have adjectives like below ( I am sure it is), doea that mean it can be used as as an adjective in a pre-noun position? Does it apply to all adjectives in both past and present participle forms?-- Often but not always.

It has been ringing. --OK

loud-ringing bell-- OK

How could you use the participles "fallen" as an adjective that pre-modifies a noun? -- The fallen hero was mourned by all.
.