are the dawn guard troops and vampires still around after you finsh the quest like will you run in to them while woundering around the woods or roads aimlessly or is it once you finsh there gone for good or at least for public viewing

If on the Dawnguard side, yes you still get vampire ambushes but the frequency is significantly reduced after finishing the questline. On the vampire side, I don't know.

Yes, as Terra Nova says, you still get the occasional ambush but the city attacks stop.

you can still run into vampires, but only at random spawn locations at night
so no more random city attacks to worry about

You can still force random vampire city attacks if you blot out the sun, but if you don't the attacks in the cities stop completely.

Unless you're bugged, city attacks stop. But the random encounters were never a part of dawnguard, simply altered by dawnguard.

That goes for both sides. Unless you go vampire and attack cities yourself, if you count that xD