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.
Well, the overarching goal of the game is to get down as close to the core as possible. To do this, you need to upgrade your gear, which you do by going through levels, getting crowns and such, which you can use to buy recipes and CE for crafting (You'll also need certain materials, but those are a minor problem by comparison.)
As far as the plot goes... well, I'm afraid that's not a major point of the game. There's certain boss levels you can go through (The Snarbolax in tier 1, the Jelly King and Roarmulous twins in tier 2, and the Firestorm Citadel in tier 3), which have recon modules at the end which give you a bit of info regarding the setting and Alpha Squadron, the recon team that went to the core and disappeared. The other bit of plot is down at the core (the end of tier 3), there's a team of NPC knights there doing research, and that's about it. Hopefully we'll get an update soon with some info about what the core actually contains, but who knows?
As far as the mineral gates thing... the type of minerals you dump into the gate will affect what kind of levels the gate contains when it comes out a day or so later. It's nothing that really affects the plot in any huge way, and it takes a ridiculous amount of minerals to even change the theme of one group of levels.
However, keep going, the gameplay becomes a lot more interesting once you get down into tier 2 and the enemies start posing an actual threat, as then you need to start using some actual tactics instead of just repeatedly pressing your mouse button.
Best Answer
As of July 30, 2013 update, materials are dropped independent of other players (stuff you see won't be seen by others).
This means you won't have to worry about not getting certain materials, all material drops (you must collect them yourself) are guaranteed to go to you.