"We are moving next to a nice city."
Translation:Egy szép város mellett haladunk.
The English sentence would be assumed to mean that we currently live somewhere else and are going to begin to live in a new house next to a pretty city, but I'm assuming that the Hungarian sentence simply means that we are in motion in that location. Correct?
There was a similar question recently where I suggested "moving along" as a good possibility for halad, especially when it means literal motion. I think that works well here: "We are moving along next to a nice city." A motion verb together with "past" might also do a good job: "We are driving past a nice city" (this suggests the city to your side - you can see it through your car window as you move along). If "driving" isn't appropriate, then "riding" or "running" or whatever would do just as well.
"Proceeding past" or "proceeding by" the city are possibilities that get the meaning right, but they sound a little formal. "Proceed" and "proceeding" have an air of school or law or other formal settings, at least to me.
No, mellett needs to come right after the thing that it's modifying. So in order to express "next to a nice city", you must have the phrase egy szép város mellett with the words in that order.
That would be moving "to" a nice city, not next to.
"Elköltözünk egy szép város mellé" would be good if you want to talk about moving your residence there.