After every battle there seems to be a grade based on how well I do. However, I can't tell what the weapon level rank does but it is consistently dropping my mission rank down a level or two. What is this based on and how can I improve it?
I generally agree with The Answer, but want to offer a few additional points. The only reason to keep AWL low (or at least under the limit specified for the level) is to avoid scoring penalties and get Platinum after-action awards. In that case, you will have to do a careful balancing act with grinding; if your AWL is too high, your score is penalized. If your AWL is too low (pushing right through all the battles with as little grinding as possible), your score is penalized TWICE, for taking too many turns and for making too many attacks. Generally speaking, if you can take care of the former, the latter happens naturally. The only thing that determines whether or not you finish on time is offensive power. One-shot-kill skill activations like "Aim-Body" and "Body Smash" or combos like multiple "Pilot Damage" activations can help. but you can't count on luck like that. So, you generally want to keep your party's skill levels one or two below the requirement; find a game FAQ to learn what the requirements (by level) are.
A good way to hold the AWL down is to always have your missile unit equip a secondary melee weapon. It will get some usage, but vastly lag behind in levels. Or, (re)introduce a dual-wielder of dissimilar weapons [Double Assault (gun+melee), Double Shot II (shotgun+MG, shotgun+flamer, rifle+plasma)], where the second weapon's skill is less. Example: party has uniform skills of 7 (B++) in their primary (single) weapon, the missile unit has a missile skill 7 and melee skill 2 (A+). (7+7+7+7+2)/5 = 6.0 Under the rounding-up factor, if any one of those levels goes up by one, the average becomes 7! But if the scoring limit for the level is 6, you are essentially doing it with level-7 weapons! And you thought math was useless after high school? :)
Leave yourself room to gain a skill level during the battle without putting yourself over. If there is a miniboss, try to have your weakest skill deliver the killing shot.
The final levels of Emma's story penalize you for AWL > 19 (E++), so there's no reason to ever exceed that, but the battles are winnable (in my experience) with AWL 12 (C+++ -> D+)!
Edit: If you're trying to keep AWL low, one more thing to watch out for, in the very first stages of Emma's storyline - DO NOT EQUIP "EXP x2" skill if you learn it! It will push her weapon skill too high too quickly, and cost you your Platinum rating. Only use an EXP-multiplying skill if/when you change a character's weapon package, and need to catch them back up to the rest of the party!