Learn Farsi: بهش زنگ زدم، ولی خونه نبود.

Behesh is the conversational, spoken form of be oo (to him/her). It attaches the preposition be (to) to the personal pronoun suffix -esh (him/her). You will hear this much more often in everyday speech than be oo.

Does zang zadam mean calling someone on the phone, or calling their name out loud?

It specifically means calling someone on the phone. The word zang literally means 'bell' or 'ring', so zang zadam translates closely to 'I rang'. If you want to say you called out someone's name, you would use a different verb, like seda zadan.

I learned 'home' as khaneh. Why is it khoone here?

Khoone is the standard spoken form of the written word khaneh. In spoken Farsi, the 'an' or 'am' sounds very often change to 'oon' or 'oom'. This is a very common pronunciation shift you will hear constantly.

Why isn't there a preposition before khoone to say he wasn't 'at' home?

In everyday Farsi, when talking about being at or going to a common destination like home, work, or class, the preposition (like dar or too) is usually dropped. Khoone nabood is the most natural way to say 'he wasn't home'.