やがる is a verb since the endings it takes is the same as the other verbs:
やがる attaches to stems; it is an affix that attaches to a verb and creates a verb.
Its meaning is to add the first person's feeling that the act was done disfavorably or turned out to be disfavorable to the first person. It is the opposite of
-てくれる, which expresses the first person's feeling that the act was favorable. Unlike
やがる is colloquial and mildly insulting.
'put a hat on someone' [Neutral]
'put a hat on someone' [Favorable]
'put a hat on someone' [Disfavorable]
In your last example, the reason
やがる is used despite the positive connotation from
よいもの is because the first person is expressing jealousy (serious or not). From the point of view of the first person, it is not a good thing that the other person made a good thing.
Traditional grammar confuses classification of words based on their meaning and classification based on the (morphological) form. It considers whatever word or morpheme that has the meaning comparable to the meanings expressed by auxiliaries in western languages (such as modaility, etc.) as 助動詞 'auxiliary verb'. However, parts of speech is a morphological (and syntactic) notion. It is a classification of words and morphemes based on their forms. It has nothing to do with the meaning. The traditional classification is inappropriate.
Furthermore, in modern analysis of Japanese, there is no such thing as 活用形. All there is is the verb stem, and an affix that attaches to it. What traditional grammar calls 活用語尾, on which 活用形 manifests, is actually the initial vowel of the affix (and/or some few sounds surrounding it). For example, traditional grammar says that the negative form verb
書かない consists of the stem
書, its 活用語尾
か and a 助動詞
ない. But notice how unsophisticated and complicated that analysis is. In modern analysis, all there is is the verb stem
kak- and the negative affix
-anai (which is itself an i-adjective).