Do you need premium to change your gun camo

No, your stats do not carry over platforms.
I can't find an official word from DICE, but I know they don't from experience.

I wish they did, I just built a PC and I'm rank 45 on the 360 :(

Sprint Jumping is the fastest method of getting from A to B, with the caveats that you become more visible and less capable of quickly killing a target thanks to the time it takes for your aim (the random spread, not your physical view) to settle down.

The part of the animation when you land from a sprinting jump moves your avatar faster forwards than the regular sprint speed, so continuously jumping while sprinting causes this part of the animation to repeat and over time this adds up to a significant bon

For proof: see this video where two people run side by side, one sprinting, the other sprint jumping. Also just spend a couple of minutes with a friend to test will show you that sprint jumping and the sprint perk are the only way to increase sprint speed, no matter the weapon held or loadout equipped.

The sprint perk raises your speed by 10%, as seen in the game data files themselves.