Are "would," "been," and "being" considered to be verbs?
Yes, they are, though 'being' could be considered as both a noun and a verb.
'Would' is a conditional verb and indicates that something will happen given the correct conditions. The root of 'would' is 'will', conjugated into the conditional tense.
'Been' is a past participle, used in the perfect or pluperfect tenses. This happens when you are indicating something in the far past, such as 'I have been'. 'Have', in this case, is an auxiliary or helping verb, while 'been' is the lexical verb - the one which carries the meaning.
'Being' is slightly more ostensible. It is a gerund, which means that it can be used as both a noun and a verb. Other gerunds are 'playing', 'showing' , and anything else ending in '-ing'. In the sentence 'I am playing', you could think of it as the speaker identifying themselves with the act of playing, as if that is who they are at that moment, in which cause 'playing' would be a noun. However, its root is a verb, and so it's both.