Translation:I am making progress.
I disagry - haladok is a physical "progression" so "making progress" is not the english way of saying it - rather you should say "I am advancing" or "moving forward"
"Haladok" isn't used for 'physical progression' exclusively. You could say: "A tanulásom halad." i.e. I am making progress in my studies.
No, not much. Halad is kinda the middle part of a continuous movement. Like, you've been moving, you are still moving now, and are like going to keep moving in the future. "Progressing" is quite close, actually.
For "to go", you'd rather use a form of megy, and for "move" there're a lot of options, depending on what exactly you mean, mostly jár or mozog.