Unless you like the way a dog tag looks, then go back for them, otherwise they are not really worth it imo.
As for the weapons you find in the campaign, those are only for the campaign. It is the campaign assignments that reward you with weapons for multiplayer, I would get all of those just so you have something better than standard equipment. Again, I really found the M249 to be invaluable for multiplayer.
There are some pretty unique advantages and disadvantages to hardcore. You only start out with about 60 health instead of 100 which means fewer shots(pretty much 1 shot anywhere from a sniper) to kill you, the enemy or a team mate. No cross hairs, meaning you have to aim. No spotting and cute little triangles above the enemy's head from across the map(though you can still spot and get the spot bonus), team mates are still marked with blue but not always, unless you have them in your sights(I try to identify by uniform, still have a few friendly fire accidents). Usually more tactics and sticking with your squad in hardcore. Like I said, I have been diggin it though, adds another bit of challenge.
When does everyone usually play? Would love to jump on and get some squad time in.
Z