> Do photons quantize and transmit the magnetic force?
The magnetic force is just an aspect of the electromagnetic force ("interaction" is the word more commonly used in quantum field theory), so yes.
> Are charged particles constantly sending a continuous stream of photons out in all directions?
Virtual photons, yes; they aren't observed directly, only their effects--what we see as the electromagnetic interaction between charged objects--are observed. At least, that's one way of (heuristically) looking at what is happening. This way of looking at it has limitations, but it can be helpful.
The magnetic force is just an aspect of the electromagnetic force ("interaction" is the word more commonly used in quantum field theory), so yes.
> Are charged particles constantly sending a continuous stream of photons out in all directions?
Virtual photons, yes; they aren't observed directly, only their effects--what we see as the electromagnetic interaction between charged objects--are observed. At least, that's one way of (heuristically) looking at what is happening. This way of looking at it has limitations, but it can be helpful.