So if it's "I rest", that seems to be "Jeg hviler mig" - "I rest myself". When it's the boy resting, why don't we need "sig" as the object of the verb?
You can use hvile with or without a reflexive pronoun. It doesn't change the meaning.
If I wanted to say 'I need to rest', would I say "Jeg behøver hviler" eller "Jeg behøver til hviler"?
You could say "Jeg behøver at hvile," "Jeg har brug for at hvile," or even "Jeg trænger til at hvile." The preposition at is used with verbs similarly to how "to" is used in English. There's an infinitives lesson that explains it.
Your last option should be "Jeg trænger til at hvile."
The words to and at are called infinitive particles (infinitivpartikel/infinitivpartikler) or infinitivemarkers (infinitivmærke/infinitivmærker)
Det var så lidt!
Forresten: Tak skal du have.
de means they – du means you (sg.)
Most languages have a formal and an informal word for you and we have it same in Danish with 'De' (formal) and 'Du' (informal). However we really don't use the formal one today. The only examples I can give are when talking directly to very old people or to one of the members of the royal family. http://programmer.tv2.dk/tillykke-margrethe/2015-04-13-dronning-margrethe-afbryder-journalist-vi-er-ikke-dus
In Danish that polite form is extremely rarely used. Sometimes even elderly people might get offended. My Danish teacher said she hopes to remember to use it if she ever happens to meet the queen. Otherwise - no.