Every level can be done with these weapons: Arc Spread, Big Laser, Lightning (the best one to get, use it on every level once you have it). Those are the main three. There are two boss levels where it helps to have slow time, levels 27 and 58. The only other weapon you may need is the rail gun for early on. But otherwise, the rest of the weapons are just fluff.
For the hard badge, you have to make it to 1839 in order for it to register. The badge says win or lose, but that really isn't accurate if you lose before 1839!
So basically, always select the mage. For all missions except control missions, just have all of your party defend in a good spot and fire offscreen with the mage. You will win easily. For control missions, stick with the mage and run around as fast as you can taking over whomever is leading at the time (or in second), but stick to a set of three or four flags, don't try to get them all. You'll win easily.
Indeedo.