「外国人登録をする時もいっしょについて」 Why ついて?

Question

I am confused by this sentence in my textbook and I hope somebody will help me.

(talking about the host family, specifically the mother/wife)
外国人登録をする時もいっしょについて来てくれました

I think this is saying the wife in the host family helped with the alien registration. But the pattern does not translate well to me.

At the time of doing the alien registration, together we...

Then I think it is strange because について is to highlight the subject, and then...

来てくれました

I appreciated her coming?

I would say:

外国人登録をする時は　お母さんといっしょに会社に行きました。助けてよかったです。

but perhaps this describes a different situation. Perhaps she met him at the office to do alien registration.

Could somebody break down the textbook sentence please? I am finding it uses odd patterns to me.

Answer 1

It's not as easy to tell in the original sentence since ついて is written in hiragana, but 付いて来る is a phrase meaning "to accompany".

外国人登録をする時もいっしょに付いて来てくれました
(My host mother) accompanied me to the alien registration.

The くれました is the -てくれる pattern which means "to do the favour of doing X". It essentially indicates that this is a favour that they did for you and that you appreciate it (much to the same effect as the 助けてよかったです in your version of the translation, but nicely packed into one sentence), and also helps to imply that the subject/topic of the sentence is you, the speaker.

Answer 2

(talking about the host family, specifically the mother/wife)
外国人登録をする時もいっしょについて来てくれました

I think it is strange because について is to highlight the subject, and

As any particle is, when について is used to mean 'about/regarding', it should be placed right after the word that it's indicating: ex. あの事件について, 次の集会について, etc.

In this case, it's NOT talking ABOUT いっしょ.

It might seem redundant, but we just often say いっしょについて行く or いっしょについてくる.

Here いっしょ becomes an adverb with the help of に.
ついて行く is used as one compound verb.

And you are right that this くれ of くれました is expressing the speaker's gratitude.