Translation:I do not walk away from the door, but from the long fence.
Walk over? I have had that translation only with át- and oda- so far.
El- usually means "away", but it's not really translatable in many cases. You should translate elsétál as "walk away" if there's only the starting point given, and as "walk" if the goal is given. (Elsétálok az állatkertbe. - I am walking to the zoo.)
I recall being told that we only split out the preverb, as a result of negation, when that negation is negating the entire action. Like: I do not walk away from the door, but I stand my ground. Here we aŕe not negating the "walking away", only what is walked away from. So, shouldn't the "elsétálok" have been left in tact?
You have it backwards. If the entire action is negated and replaced, you leave the prefix on the verb. Splitting it off is the default move when you're making a negation.
- Nem sétálok el. - I am not walking away.
- Nem elsétálok, hanem megmaradok itt. - I'm not walking away but staying here.