I've played through the first six missions of Ground Control, and I have a few questions that the manual does not adequately answer.
1. What is the importance of the medals a squad earns during a mission? Do they simply keep track of a squad's stats, or is there more to them? What are the different kinds of medals you can get?
2. I know that a squad's increasing a squad's experience level improves the squad's stats, but I don't quite grasp how a squad gains experience. Is it tied to the number of medals they have?
3. How many experience levels are there? I've gone from "Inexperienced" to "Blood Veteran," but I'd like to know what ranks there are beyond that.

Yeah I'd like to know these things too.
To be honest, I haven't seen any kind of improvement in a squad's stats, even though they gained medals and their rank increased.

1. Medals are basically rewards for particulary hard-working squads. If I recall correctly, a squad with lots of confirmed kills in a single mission is awarded with the "Field excellence" medal, one that destroyed lots of buildings the "Destroyer" medal and one that engaged lots of enemies directly and took damage the "Aggressive" medal. Additionally, every squad is awarded with the "Combat Star" after its fifth active mission.
A squad that did not survive a mission (K.I.A.) still may recieve a medal/medals posthumously.
2. The more enemies a squad kills and/or the more buildings it destroys, the faster it gains experience. The medals are merely a nice side effect.
3. There are five expereience levels for Crays and Dawnies respectively:
Crayven Corporation - Inexperienced > Experienced > Veteran > Blood Veteran > Elite
Order of the New Dawn - Unproven > Faithful > Priestly > Battle Brother > Sunbringer

The stats that you refer to (Firepower, Armour and Speed) are not affected by a squad's experience level - they are technical limitations that can only be changed in the "Squad Configuration" menu (Balanced, Offensive, Defensive and Speed).
You have to closely observe a squad's performance to realise that an experienced squad...
- has a higher hit percentage (Accuracy)
- is less likely to be detected when moving within an enemy's view range (Stealth)
- is more likely to spot hidden/stealthy enemy units (Perception)
- may inflict greater damage on an enemy by knowing possible weak spots (Damage)