Do I need a weight distribution hitch?

My trailer is 6k-7k (depending on toys) with more tongue weight than what you've got. I do not use a WD hitch.

I had the same discussion about my 3/4 ton truck and the trailer weighted at 6,500 pounds empty. Most members said I did not need it. However, I negotiated with the dealer to include one for free. I towed both ways and I noticed no difference. Sway controller will be a benefit though.

No, you need all of the 510 is needed on the rear axle.

WD hitch is there to balance or shift part of the load to the front axle. Weights when fully loaded will tell the tale.

I doubt it. Looks like you are fine. As long as your sway control works.