Let me begin by stating that an adverb is a modifier and that there are different kinds of modifiers.
An adjective attribute can be either before or after its main word, i.e. bela domo or domo bela. The former way is nowadays more common.
Prepositions and particles go before their main word, i.e. antaŭ la domo, ankaŭ mi. The exception is the particle mem which goes after its main word, i.e. ni mem.
When it comes to adverbs, the recommended position is the same as with most modifiers, before the expression it modifies. The modified expression can vary from a single word (which even can be another adverb) to a whole subclause (see PMEG for further info).
Note, that when there are two adverbs modifying the same expression, the expression is usually a single word and the adverbs surround it. For instance ĉu vi ofte kuras rapide?
Ili multe ŝatas pastaĵojn.
Here multe modifies the verb, so it intensifies the liking.
Ili ŝatas multe pastaĵojn.
Here multe stands before the object, but does not modify it, because there is no da, i.e. the example phrase does not mean the same as Ili ŝatas multe da pastaĵoj. Therefore it must modify the preceding verb.
When it comes to the intensity, the speaker's and listener's native languages play a role. If after a verb is a normal, neutral position for an adverb in the speaker's language, the speaker's intention may be "normally intensified". If not, the intention might be "specially emphasised intensity". Add to this the listener's perception, and you realise that the perceived nuance might not be the same as the intended. In practise the nuance difference is mostly negligible.
Ili ŝatas pastaĵojn multe.
The last position indeed relates to all the preceding. As Joop Eggen already pointed out, multe in this position can be perceived to refer to time, i.e.≈ Ili ofte ŝatas pastaĵojn, but this is a special trait of multe. Should you use tre, *Ili ŝatas pastaĵojn tre* would be considered erroneous use.
There is some leeway where to put an adverb. Before the expression it modifies is suitable for all kinds of expressions (e.g. before a subclause) and likely perceived by many as the most Esperanto-ish style. After a single main word is also common and in many cases equally fine. Any other position is highly emphasising, perhaps triggers a special meaning of the adverb or is bad style, if not outright wrong.
Even if PMEG recommends using and some respected authors use commas to associate an adverb to a particular part of a sentence, I would avoid this kind of use. Not counting mathematics the comma rules in Esperanto are not steadfast and speakers of different languages can have varying views about their use.