Learn Farsi: از همه‌ی رنگ‌های این فرش سنتی به جز سفید خوشم می‌آید. - I like all the colors of this traditional carpet except white.

No, be joz (except) does not take an ezafe. You just place the noun or pronoun directly after it, giving us be joz sefid (except white). This makes it a bit easier to use than be jâ-ye (instead of), which does require an ezafe.

Why does the sentence start with az when there is no "from" in the English translation?

The word az (from) works together with khosham mi-âyad at the end of the sentence to mean "I like". In Farsi, the structure for liking something is az [X] khosham mi-âyad (literally: pleasantness comes to me from [X]). The az goes at the very beginning of the whole block of things you like, which in this case is "all the colors of this traditional carpet except white".

How do we know who is doing the liking in khosham mi-âyad?

The person doing the liking is shown by the attached pronoun -am (my / to me) on the word khosh. If you wanted to say "he likes", it would change to khoshash mi-âyad (his pleasantness comes). The actual verb mi-âyad (it comes) stays in the "it" form because the subject of the verb is technically the "pleasantness", not you!

How does the linking ezafe work in the phrase hame-ye rang-hâ-ye in farsh-e sonnati?

The ezafe (the "e" or "ye" sounds connecting the words) acts like the English word "of" or links a noun to its adjective. Piece by piece, it links the phrase together: all of () the colors of () this carpet which is () traditional ().