In principle, deep is an adjective (modifies nouns), and deeply is an adverb (modifies verbs). In practice,
He cut deep into something. (about 2190 results in Google Books)
He cut deeply into it. (1030)
By which you can see that native speakers tend to ignore that principle when the verb is being used "actively". But when it's just a "passive" past participle, we invariably follow the rules...
He was deep hurt (1)
He was deep wounded (4)
He was deeply hurt (55000)
He was deeply wounded (20400)
Also note that in some constructions that might superficially look like "adverbial" usages, the word is actually part of an adjectival "modifier of location". We use the simpler form there as well...
He walked deeply into the forest. (38)
He walked deep into the forest. (7390)
In OP's example, deep is used adjectivally as per the above (it's telling us where the chicken was buried). It would be perfectly grammatical - if somewhat less common - to use deeply there (in which case the emphasis would be on how she buried it, not where).
This use of superficially adjectival forms in adverbial contexts is quite common in English. I can't think of an example where the difference between deep/deeply would be perceived as changing the meaning, but this certainly occurs with other words. For example...
1:He acts bad.
2:He acts badly.
...where in the absence of any other contextual information, we naturally tend to assume that #1 means he acts like a bad person, whereas #2 is likely to be interpreted as he's not good at acting.
It's not a pronoun because it doesn't take the place of a noun - not by itself anyway. No one/nobody/nothing/nowhere would be some of the "pronoun forms" of no.
No generally modifies a noun so that makes it technically an adjective. When modifying a verb or adjective not is used.
No is also a typically a determiner - which can be considered a special type of adjective - because it can occupy the "spot" where an article or word like this/that/these/those would go.
Best Answer
You can also understand that sentence like this:
They dug the well (to be | so it was) deep.
in which case deep is a characteristic of the well, not a manner of digging.
Compare:
P.S. Consider the following. The parentheses around to be indicates that those words could be used in an actual statement but may be omitted without change of meaning.
deep is a characteristic of the (intended) well, not of the wanting itself.
Transitive dig here is directed (volitional or intentional) action.