"He does not like anything."
Translation:הוא לא אוהב כלום.
If you say הוא לא אוהב כל דבר what you mean is that he's very picky and likes only very specific things.
Yes. But it's not used in the negative. We use כלום or שום דבר, which literally mean "very little", and so the sentence becomes literally "He does not like even very little" or something like that. But we don't think about it like that, it's a fixed expression.
Once again the missing "את" is confusing me, but this time as to like/to love is a verb, shouldn't "את" be there before the object so we get "הוא לא אוהב את שום דבר" or "הוא לא אוהב את כלום" or is "אוהב" seen as an adjective here and not the present tense of "לאהוב"? Am I missing something?
Ohev is not an adjective, it is a verb. The reason there is no et is because et is only used when there is a direct object, and "anything" isn't a direct object. If it was "He doesn't like that", it would require an et as in הוא לא אוהב את זה.
If it helps: direct object basically means the object needs to be specific. The cat. Not any cat. But this specific cat. Here it would be direct object and needs את. But nothing/anything is not specific. Could be literally anything. So it is not a direct object and does not need את.