てくる is divided to て(and) 来る(come).
So, Basically, てくる means some action and come (to me, here or back here).
If directly translated,
My mother bought a shirt and came here/back here with the shirt. (not sure for who)
Just 買った means "My mother bought a shirt" (not sure where she is now and not sure for who).
くれてきた。 is something wrong.
Instead, we use きてくれた
My mother bought a shirt and came to me and gave it to me. (Directly translated)
My mother bought a shirt for me. (Translated naturally)
買って(buy) + きて(come) + くれた(give)
So, the order of 買って + くれて + きた is wrong.
You cannot give something to someone directly without coming.
Also, 読んで + くれて + きた is wrong because 読んでくれた(read a book for me) and come, is weird, right?
By the way, the correct order, 読んで + きて + くれた is very strange, but can be used in a rare situation.
(I have to read a book and write an essay as homework, but) my mother read the book in a library and came back here (and maybe writes the essay for me).
This is a very difficult point in Japanese, but 送ってきた is not 送って(sent and) きた(came here).
This means just "sent it to you".
I guess the reason is as below.
When you write a letter to someone and send it, Japanese thought he/she is in the letter or his/her soul/feeling is in the letter. It means that you/your feeling literally comes to me riding on the letter like a train, bus or airplane.
So, おくってきた can be 送って(you sent it to me and) きた(your soul/feeling comes to me)
Now, we use 送ってきた as "sent it to me/you" although it is not a letter, but it is an email or even a gift box.
However, 送ってきた also can be "send + come here".
For example, You asked someone (her) to send your mail to someone, and then she went to a post office and came back.
She would say "I sent it to him" translated as "送ってきた".
This is "send + come". I think this case is occasional.
So, "送ってきた" can have both nuances. "send it to someone and receive" or "send + come here"
You need to judge from the context...
So, let me answer your three questions.
can any verb be used?
Basically, you can use with any verb, but some words should be weird.
For example, 来る(come) means already "come", so if you use てくる with 来る, it is "来てくる", that is come and come. It is weird.
Can use with abstract verb.
As I mentioned "読んできてくれた", you can use though it depends on situation.
So, 君の考えを聞いてきた can be used.
Whom did someone send it? in the case that "田中さんが手紙を送ってきた"
It should be the person who said "田中さんが手紙を送ってきた", that is 私.
However, in the following situation like question sentence, the object is not 私:
A: I received a letter from Tanaka-san.
B: Tanaka-san send it to you!?
Anyway, the object should be obvious in the context.
I am sorry that this answer is so long, but it is difficult and good questions.
Note: See the comments to get more detailed answer