Translation:Mom, I'm off.
Because that's the literal meaning of the word. Now, just like はじめまして and many other greetings, the literal meaning isn't necessarily preserved, something to be wary of when translating.
Your suggestion sounds more like a statement of what you'll be doing, rather than the departing greeting that this phrase is.
Just adding to @kzulu92 's answer, 行ってきます is indeed a form of 行く. Specifically, it's the て-form plus 来ます (きます = "to come").
This grammar structure is relatively common, though it's more often used with other verbs, for example: 「取ってきます」(とってきます) means "I'll go and get it" (with the implication being that you'll bring it back) since 取る means "to pick up/take".
This of it as "I'm going out, I'll be back".
行く is to go. 来る is to come. So you'll go, then come back. 行って is the て form of 行く and can be used many ways depending on what follows. But this one is somewhat of a set phrase, I believe.
お母さん is the polite form, 母 is the plain form. You would use the plain when referring to your mother to an outgroup member, but not when you're within the family in-group. That's because using the honorific can sometimes be seen as putting down another person's mother. お母さん would be a commonly expected level of respect when you're just talking to your immediate family, especially if it's a formal household.
I think it should be accepted, but to be really pedantic, there is a subtle difference, at least in English, in that "going out" usually has the connotation of being for some leisure activity. Perhaps this is only true in my native Australia, but you would seldom hear someone say "I'm going out" before leaving for work, for example.
In that vein, I'd be more inclined to translate "I'm going out" to もうでかけるね？, rather than いってきます, but I'm aware there isn't a hard line separating the two.
Good question. I'm not a native speaker, but my understanding is that the nuance of 行ってきますか (which is a distinctly different use case from the greeting) is more like "(so) are you going to go (to the thing you were undecided about) (which I'm not going to)?"
"Are you leaving" sounds more like 出掛けますか【でかけますか】which is more akin to asking if someone is heading out (in general, to an unspecified location) of the house.
No, 行ってきます as a greeting is reserved for when you are leaving to go to a different location from a place you are very likely to come back to, e.g. going out of the house or getting on a plane (and leaving the other person behind).
While you could argue that "your bedroom" is a different location from "the living room", for example, going to your room to sleep has stronger connotations of "returning" rather than "leaving", so 行ってきます isn't really applicable in that situation. Or rather, おやすみ(なさい) is far more applicable.