You can swap the emblem and shader to your alt characters, but have to reach rank 4 with each class to buy the Class Item. If your alts are the same class, then you can swap that class item as well.
Also, holy god buy the Tempered buff. It's enormous and worth buying, even on day 1.
One more tip: IB bounties don't reset until you log in after the rollover (1 am PST). So if you're one kill away from finishing your auto rifle headshot bounty, and it's after 1 am, just keep playing until you do it. Turn it in, log out, log back in, and you can pick up a fresh auto rifle headshot bounty right there. In contrast, if you log out with 1 kill to go, log back in, get the kill and turn it in, you won't be able to pick up the bounty again until after the next rollover.