grammar - Understanding てくる in these sentences

Coming across lines such as (two separate contexts):

「ご飯をよそってきて食べなよ」

and

「そんなに驚かないで。頭を冷やすために下でお茶でも淹れてくるよ」

I'm having trouble understanding the usage of 来る in both of these sentences.

I was told that the former sentence, 来る functions as an auxiliary (which makes sense in the surrounding context, since it isn't indicated that the person being told needed to move between different locations), but am then a little confused what it specifically functions as (besides perhaps,"to come to serving their rice", "to start serving their rice").

The second sentence seems much more clear with 来る functioning as a full verb, "to brew tea and return", "to go brew tea", but with how I understand the first line, I started getting second doubts here and thought that it could function as an auxiliary also, "to come to brew tea", "to start brewing tea."

Is there a general rule of thumb for particular verbs used with てくる, or is it solely reliant on context? Any clarification would be greatly appreciated, as well as correcting any misunderstandings I might have with 来る.

Answer

「ご飯{はん}をよそってきて食{た}べなよ。」

「そんなに驚{おどろ}かないで。頭{あたま}を冷{ひや}やすために下{した}でお茶{ちゃ}でも淹{い}れてくるよ。」

First of all, 「くる」 in both sentences is a subsidiary verb, which is why it is written in kana in both. When used as a "regular" verb, it will certainly be written using the kanji as 「来{く}る」, 「来{き}て」 etc. because it is such a simple kanji that even the 1st-graders can read and write.

「Verb in てｰform + くる」　means:

"to (verb) and come back (to where one was)"

In other words, the place where one can perform the action described by the verb is at least a few steps away from where one is.

The sentences in question mean, respectively,:

"Why dontcha go scoop some rice (and come back here) and eat it."

"No panic! I'll go make some tea downstairs (and return) so it'll cool you down."

Both the rice and the tea are at least a few steps away, correct?

If the rice were located right where one was like right on the table where one was sitting, one would not say the first sentence.　Why not? That is because no "returning" would be needed. One would simply say without a 「きて」:

「ご飯をよそって食べなよ。」

Similarly, if the tea leaves and (hot) water were right where one was, the second sentence would be:

「頭を冷やすためにお茶でも淹れるよ。」

as "going downstairs (and returning upstairs)" would be unnecessary. I dropped the 「下で」 and 「くる」.

Off the top of my head, the only verbs that the subsidiary verb 「いく」 or 「くる」 could not be attached to would be 「ある」, 「いる」 and 「来る」. There might be more, so please add if someone can think of other verbs.

It is correct to say 「行ってくる」, which is why we say 「行ってきます。」 when leaving home for work or school.