1) The verb 약속하다 needs a noun in the form of Verb + -기. The particle -로 tightly ties a noun to a verb. But I just can't figure out what its purpose is with the verb 약속하다.
우리 아빠가 (나에게) 이메일을 보내기로 약속했다.
The expression "-기로 하다" is usually used when saying the facts about the something determined, scheduled ,or promised.
Sometimes, to clarify the meaning of it, the verb 하다 can be take place of another verbs such as 약속하다, 결정하다, or 결심하다
The particle "-로" is usually used to show the direction of something such as motion, progress, and it is often used with a decision, promise or choice. For example,
What would you like to have for dessert? Do you want coffee or tea?
후식은 뭘로 할래요? 커피로 할래요? 아니면 홍차로?
I would like to have a coffee.
저는 커피로 주세요.
2) Is there any difference between the following sentences? The last one uses "quoting", but in effect, they are all using quoting really.
우리 아빠가 이메일을 보내기로 약속했다. (with expression "-기로 약속하다")
우리 아빠가 이메일을 보낼 것을 약속했다. (promise to do something)
우리 아빠가 이메일을 보낸다고 약속했다. (promise that ...)
All sentences above have the same meaning.