"I don't want beer."
Translation:Nem kérek sört.
"Kérek" is more polite and in most situations that is the better. But yes, "akarok" is literally fits here, and in some situations it is acceptable, too. In very informal company, or among people rated lot lower than you the "akarok" is the right one.
Can someone explain to me why sometimes the direct object, "sört" in this case, can go before the verb and not others?
If you put sört before the verb, you change the meaning of the sentence.
Nem kérek sört. is I don't want beer. And it could be that i don't want anything else either. Nem sört kérek. is It is beer that I want but something else.
If you put "sört" in different places then you change the emphasis of the sentence. I don't know what you have in mind when you say "not others." Can you give me an example?
My response was "Nem sört kérek." which I thought was correct but it did not accept it.
It's correct but it puts the emphasis on "sört". It means "It's not beer that I want (implied: but something else)"
Yes, that has a slightly different meaning, it is more polite but meanwhile it suggests uncertainity, too. (It depends on the context whether the polite or the uncertain overtone is stronger.) "Nem szeretnék sört" would be "I wouldn't like beer", and it also implies that you would like to have something else, like "nem szeretnék sört, de kávét igen!" (I wouldn't like beer but coffee). For uncertainity part, it is better to explain a dialogue:
"Come on, let's have a beer!"
"I wouldn't like beer."
"Okay, what would you like instead?"
"Dunno. An ice cream?"
– Gyere, igyunk egy sört!
Note this is not word by word but idiomatic translation!
– Nem szeretnék sört.
– Oké, mit szeretnél inkább?
Lit. Oké, mit szeretnél helyette? but this is rare in spoken language
– Nem'tom. Fagyit...
Nem'tom = Nem tudom: rather informal. Note that that type of hesitation that is often written in English with question mark at the end of a statement, we express with ellipsis. Also dialogues are written with n-dash ("gondolatjel") in front, without quotation marks. In most cases n-dash is followed by a space, but this sometimes omitted.