As seen from my attachment (as I couldn't draw it with the program), tree 1 represents my proposal of how the tree would start from top down.
You can draw the tree top down or bottom up. Either way, it will be the same tree. Your instructor might suggest one method or the other, but that's just to help you learn how to draw the trees.
I propose also, that [quite fond of his cats] as an AP, as seen in the second tree.
Correct.
However, there is no syntactic rule (as those we discussed in class) that seem to provide for A V' --> V AP or along those lines.
Some verbs take nouns as complements (objects). Others take adjectives. So you do need a rule like this:
VP --> V AP
(Exactly how the structure works, such as with V' etc., may vary based on what your instructor suggests.)
In addition, I suppose it is crucial to figure out the lexical category of the word "is". I've consulted many, and they deem it to be a verb, but I'm thinking that it should be an inflection instead?
That's a little complicated. Generally speaking, it's a verb. But it's one of the verbs that can raise to INFL if needed, along with other auxiliaries. So you can either simplify the tree and omit the IP, or you can include movement (from V to I), or you can add a null V head where a verb would go.
(Alternatively you could introduce a new kind of rule along the lines of this: IP --> I AP, but that seems a little strange to me and is inconsistent with most analyses where there's [almost??] always a VP.)
