Tie light/large/high stuff down- It's common sense and good safety practice.
Tailgate thing is been touched on.
It's easy to make the changes to make the trailgate more "permanent"
You don't NEED weight in the bed. I drove a super charged Dodge Dakota 4x4 and had no problems at all with no weight- in the same area that you live in.
It will probably give you a better feeling-a little more sticky I guess- in the rear end of the truck- but you can control the truck without it...You'll be fine either way-just gotta decide what you like better.
I have a feeling your old Dodge weighed a bit more than these trucks, or at the very least had more of the weight over the rear wheels due to it having a metal bed.
Trucks in general are light in the rear where the actual drive wheels are, and these trucks are worse due to the composite bed. I would not drive a 2wd version of this truck in snow without weight in the bed or better tires. Almost lost mine 2 weeks after delivery on a light dusting of snow. Then again, it could've been the POS Rugged Fails it came with. Sand in the back made it a little better, but I started automatically engaging 4wd as soon the roads turned white. Still did sometimes with the new tires after the trauma from the last experience. Nothing like almost seeing your 2 week old truck drive off a cliff.
I cut the $15 carpet to be a tight fit, and no problem with it moving at all.
By the way, you asked about adding weight in the bed during the winter. I don't,I prefer to drive carefully, not drive at all in bad weather, or use 4WD when necessary. If you decide to add weight, consider whether you want the item(s) hitting you in the back of the head in a possible collision. I won't use cement blocks for weight in the back for this reason. Sand bags maybe.
With that said, first thing I'd do is dump the stock tires before the snow flies. Save them for spring through fall if you want, but for take them off before winter! My Bridgestone Rugged Trails worked fine in deep snow, 4WD, but couldn't handle light stuff on the roads. Tread packed full, wouldn't clean out, and suddenly they became giant snow balls rolling on snow. Talk about lack of traction.