Is the word order "Akarok szép városokat látni" permissible here?
Maybe, maybe not. It sounds like an answer to the question "Do you want to see pretty towns?". Answer: "(Yes,) I (do) want to see pretty towns."
The sentence above is more like an answer to the question "What do you want to see?".
what's wrong with: "Szep varosokat szeretem latni"?
what's wrong with: "Szép városokat kérek látni"?
it is more like " I would like to"
What's the "ni" stem in verbs mean?
see - lát ; to see - látni (I see towns. - Városokat látok. ; I want to see towns. - Városokat akarok látni.)