So, I've been reading through the rules again after my first session as GM with my new group, and I've still got a few questions. First, what's a good amount of credits with which to reward my players, representing their Alliance paycheck? Second, is it reasonable for me as GM to make melee attacks an opposed check? Third, my Players spent an inordinate amount of time making medicine checks; at one point, out of combat, they used three consecutive rounds/turns to heal up. Is this rules-legal?

Any advice?

The advice is to keep players desperate for money.

Melee is structured the same as Ranged attacks so if you make it an opposed check why would anyone melee if they can just use Ranged weapons and not bother.

You get one Medicine check at the end of the encounter.

On credits, most of the Edge adventures have around 500-2000 credits per PC per adventure. The Dangerous Covenants book also has a handy chart in the back for typical pay on Hired Gun jobs, which may fit well with soldier's pay. But it's largely up to how quickly you want your PCs to be able to acquire things.

What 2P51 said on melee. The rules are the way they are for a reason. Play through the system for a long while before directly changing anything.

On medicine, to clarify what 2P51 said, each PC can have one Medicine check performed on them to heal wounds once per encounter, because as the book says, first-aid can only do so much until you're back at an infirmary/medbay. So the Medic can take a few rounds making checks on multiple people, but only once per PC. Same goes for crits. Some good conversation on this over on the Edge forums: http://community.fantasyflightgames.com/index.php?/topic/109549-heal-checks/