Adverb

No. An adverb is a single word, not a phrase. It a word that describes a verb (an adjective describes a noun).

Consider:

I quickly ran to get my gun.

Adverb = quickly
verb = ran

Sorry I don't understand this part of your question.

Sorry,i didn't make it clear."They nodded TO ACKNOWLEDGE HIS SUGGESTION". This phrase is a verbal, infinitive phrase used as adverb. Because the phrase answers the question "why"? What about this sentence?The suspect was forced to pay the bill. Is the infinitive phrase used as adverb?

Yes, the sentence could be an answer to a WHY question, such a why did the suspect pay the hospital bill. He was forced to do so.
Why does the suspect have no money? He was forced to pay the hospital bill.

An adverb is a single word, not a phrase. There is no adverb in the capital lettered phrase. To Pay(verb) the Victim(noun) Hospital - (adjective in this case - normally a noun) Bill - noun.

I think so, because it says why they nodded.

But to be honest I had to read this to even understand what an infinitive phrase was!
http://www.chompchomp.com/terms/infinitivephrase.htm

But there are adverbial phrases, Hingehead, Ceili.

more importantly; He did it in an aggressive/unwieldy/difficult fashion

See:

http://web2.uvcs.uvic.ca/elc/studyzone/410/grammar/advphr.htm