て form can't be considered as a noun, grammatically. It might look like it fits as an explanation for those patterns, but it doesn't in a lot of others. There is a verb conjugation often used to form nouns, and it's the masu-stem (e.g. the masu form minus the "masu").
頼む (to request)
頼み (a request) - you can say, e.g. 頼みがある (頼んでがある would be ungrammatical).
(note - this is not "productive", so you can't just noun every verb you run across. But it is common).
Incidentally, there are quite a few forms where you can insert て、も、なんて and other things between the て and what follows for contrast/emphasis, etc, but て would never be followed by を・が・に
てくる・ていく are easier once you understand that くる・いく work the same way for both distance and time.
くる = coming towards the speaker (from past until now, from there to here)
いく = going away from the speaker (from now onwards, from here to there).
With てくる・ていく, I think of a change over time, not sudden.
ている is more complicated because depending on the verb and context it can be an ongoing action (I am eating), a state (I am married), an ongoing habitual action (I study Japanese), etc. But generally it refers to the present, and doesn't imply anything about the state before or after.
e.g. if you say you "分かっている" something, it just means you understand it now. If you "分かってきた", it used to be that you didn't understand, but over time you have come to understand it.
The problem with trying to understand the "why" is that to really get into that stuff you have to first learn enough Japanese to read linguistics papers. While interesting trivia, it's not a good use of time at the beginner stage.