There isn’t any hard and fast rules but you can go by rules of thumb and get a general idea of the gist of things…
For the first question, many jukugo words will be on’yomi but definitely no small amount are kun’yomi, but you might find patterns in which words use kun. Like, a lot of words that use body parts will use kun’yomi, so 手首 is “hand” and “neck” and is read てくび, for example; both kun’yomi. But this isn’t hard and fast either, like 歌手 means singer (in this instance and instances like this, the hand kanji sort of functions as “person performing the act of”) and is read かしゅ, using on’yomi. There are a couple of words like this using this exact function where the readings are all on’yomi, ie 投手 (pitcher), 助手 (assistant), 選手 (pro athlete) so as you can see there are exceptions to the exceptions. This all seems overwhelming but the light at the end of the tunnel is that as you learn, I promise you will get a feel for what the reading should be. You’ll start to guess the readings of the vocabulary before you see them, and a lot of the time you will be right. It just takes time.
As for the second question, it depends, and this is where knowing grammar becomes a big deal. In 食べる the hiragana are part of the word “to eat”, so they are a piece of the word. You can also have 折り紙 which is made up of the noun (masu stem?) version of “to fold”, 折る, and paper 紙, and is one word - origami (folding paper, folded paper etc, hopefully you know this word?). and then there are particles, so you can have 私の電話 where the attributive particle の is telling you how “I” (私) is related to “phone” (電話) - “my phone”.
The point is that there really isn’t a cheat sheet, but it gets easier the more you know.
