What exactly do Iron Lockboxes contain

At Heat Lvl 5 and 10 your weapons get a small but meaningful bonus, and yes if you took the time to look at a Lvl 1 item you see the blue bar and some dark blueish bar after that. Each level fills that up a bit. For crafting the higher tier items eg.Tempered Calibur it need the standard stuff plus a Calibur. An Acended Calibur requires a lvl 5+ Tempered Calibur. A rule of thumb is if you're not at the highest tier of gear and your stuff is lvl 10 get new stuff.

You do not get the mission rewards and prestige from replayed missions, however you do get the money and items you found within the missions. Thus, replaying missions may not be a way to profit from selling the rewards, but it makes farming Materials, Tokens and Crowns much easier by simply being able to make a pure Snarbolax or Jelly King run for example.