Translation:Even the king sometimes needs a beer.
Asked my Dutch friend:
First one is more classy/formal, second one is more common, but might also come over as more "needy". Heeft ... behoefte is more like " i would like to", heeft ... nodig more like "i need/want" (more needy due to bluntness)
I took French during high school, and hearing this reminded me of the difference between "Je voudrais" (I would like) and "Je veux" (I want)
"behoefte" is more like a "need" that you don't actually need, e.g. the 'needs' in a relationship which really indicates the things you wish to get out of a relstionship, or 'I really need a beer' is used to mean "I have a strong desire for a beer", in those cases you'd use 'behoefte'. An interesting exception might be "je(/jouw)/m'n(/mijn)/z'n(/zijn)/d'r(/haar) behoefte doen" which means "you/me/him/her taking a shit"
"Nodig" is more used for things you actually need, "ik heb water nodig" ("I need water" "ze heeft nog bloem nodig voor het recept" ("she still needs flower for the recipe" although it is still sometimes used in places where you dont really need it, but then it often is paired with "even" (so "even nodig") E.g. "Een pilsje, dat had ik echt even nodig" ("A pilsner, I really needed that right now" "Even" sort of means 'for a moment' (although it also means 'even' as in 'even or odd') but I find it difficult to give a direct translation, and it's more used as a figure of speech in this case.
I'd translate "Even the king..." as "Zelfs de koning...". My guess is that the Dutch sentence was the source, and then "Ook de koning..." might be idiomatically translated as "Even the king...", which also mostly conserves the original word order. I'd probably go for the more literal translation of "The king ..., too" or something with 'also' or 'as well'. To me, 'even'/'zelfs' implies some sort of extra-special-ness to the occasion that 'also'/'ook' does not.
It shouldn't, I'm afraid. Putting the adverb after the verb is a very marked usage. People would have no issue understanding it, but it would mark the speaker or writer out as a non-native speaker. If you placed it before the verb or at the end of the sentence, it would be much more idiomatic.