Native English speaker here.
When you're using "both" or "both of" generally, they are synonymous:
-"Both my students passed the exam." ✔ Correct -"Both of my students passed the exam." ✔ Correct
Both (no pun intended) of these sentences mean that you have only two students, and two of those two passed the exam. Sometimes one may feel more natural to use, and sometimes the other will.
The only exception where you have to use one or the other is when using pronouns, such as:
-"Both of us passed the exam." ✔ Correct -"Both us passed the exam." X Wrong
The exception is when the pronoun in question is an object, then you can put the pronoun before the word "both" and be alright:
"I saw them both." ✔ Correct "I saw both of them." ✔ Correct "I saw them of both." X Wrong
"We would like both books" is unfortunately not accepted. In most of the previous lessons want and would like can be used interchangeably. Please correct it here too - I sometimes feel as if my English were getting somewhat, er, impolite after practising too much Hebrew with Duo :)
I fear a real צַבָּר would not use such a level of polite language. I suppose אֲנַ֫חְנוּ רוֹצִים is already the improved version of תֵּן לָ֫נוּ! Some languages make very much ado about the right way to address someone (think of Korean), others are rather brusquely in our ears, amongst them Hebrew.
I do hear occasionally הייתי רוצה, and רציתי לבקש, even from Sabras... Depends on the speaker and the social context.
I wonder if using the past tense (in two forms!) for politeness is just influence from European languages, or is there something inherent / psychological in effect.