"There are many small mushrooms growing in the tree."
Translation:Puussa kasvaa monta pientä sientä.
This is the way to say it in normal speech. One could also say "Monta pientä sientä kasvaa puussa.", but it maybe emphasizes the tree more than the mushrooms. If one wants to get really poetic, then the word order can vary more, e.g. "Puussa kasvaa sientä - monta pientä." (to make it rhyme) but nobody actually speaks that way.
Yes, it's because of "monta", just like with numerals other than one. The rest of the phrase is not in plural though but in singular partitive.
Sing. nom.: pieni sieni
Sing. part.: pientä sientä
Plur. nom.: pienet sienet
Plur. part.: pieniä sieniä
"Sienet kasvavat metsässä."
"Kolme sientä kasvaa metsässä."
Two finite verbs within the same clause is not grammatical in Finnish nor English. A finite verb expresses things like tense, person, number, and so on, an example of which is "grows" since it expresses present tense third person singular. A nonfinite verb doesn't express those things and it can be in a verb phrase with any combination of tense, person, and number, an example of which is "growing". "On" and "kasvavat" are both finite because "on" expresses indicative mood present tense third person singular, whereas "kasvavat" expresses indicative mood present tense third person plural. Using those verbs in the same clause is comparable to saying "is grows".