"The boy asked his dad to read a book."
と has many functions, the one being used here is to encapsulate the phrase before it, either as a spoken phrase, a thought, or a concept. Think of it as putting either quotation marks or parentheses around the previous words.
男の子はお父さんに本を読んでとたのみました。Leave out the longer dialogue and you have 本を読んでとたのみました。
'ほんを読んで' と たのみました。He asked (of his father) , 'Read a book'.
Since there are no actual quotation marks, however, it would be better phrased as translated by Duo.
The pattern is [dictionary verb form / verb ない form] + [ように].
The よう and に shouldn't be seen independently, but as part of a set.
ように can be used with different expressions that are somewhat varying in context and intent. Below are a couple of links, but I suggest you peruse a few more to get more familiar with the various usages:
Some of these questions use the informal form for verbs and some use the て form in regards to verbs for the と particle. Which is it? when I was in class, I was taught the former. I'm pretty sure よむ should be used here, though I only learned about the particle in regards to いう and 思う.
I kinda have an inkling, that as it is a request, that is what the て form is also used for, but at the same time, たのむ is expressing that anyway. Is it purposefully redundant when requesting something? I can't think of any other way to put the て request form in the past tense other than like this.
Okay so there are roughly four basic parts, three you should be decently familiar with.
男の子は just says it's the boy doing the asking. お父さんに means he's asking his father. たのみました is just the past tense of "to ask".
The last part I assume is what's tripping you up because it acts like a mini sentence. 本を読んで means "read a book" (as an instruction". The particle と turns this into somewhat of a quote, so it makes this mini sentence into what the boy is asking his father.
You've got it a little backwards, the に is attached to お父さん, not to 本.
The other thing is that "に" can be translated as "inside" kinda but also means something along the lines of "to" (for example, in the sentence "学校に行きました", "I went to school", the に means "to"). That's more-or-less how it's working in this sentence. The boy is asking "to" (I realise, you don't say it like that in English) his dad.