Weapon stats do carry over, the numbers like 300 damage don't. For example, if a weapon has low impact, that's basically low damage in pvp. A big stat that's important on primary weapons is stability, more stability means less kick, less kick means more headshots, more headshots mean a faster kill. My first few pulse rifles and auto rifles felt fine to me, then once I unlocked rare primaries like this auto rifle I got that had more stability, I was lining up headshots even more consistent than my previous guns and I was killing people nearly twice as fast. It was also a crucible specific unlock, I think the crucible specific weapons are the best for when it comes to the crucible, but not entirely for single player. I've seen crazier primary weapons with insane stats for single player/co op, but poor stability which makes them not so great for the crucible.

I'm not sure about that. I do think that intellect and strength and the other stat carry over. I also know armor and weapon abilities carry over such as: melee faster, reload faster, etc.