As the attached screen shots show. A charachter has LOTS of first aid kits, but the game
is Refusing to allow me to use them, stating that the charachter "has no medical kit".

Okay. Never mind.

The game makes a distinction between first aid packs and medical packs.
Be wounded badly enough, and the first aid kits no longer cut it.

This game is Waaaaaaayyy too hard.

The small ones are first aid kits. The big ones are medical kits. You can use either to bandage mercs in the field but if you want to restore their health permanently, you can only use the big medical kits.

Resting restores health too, but slowly. And imo it should restore very slowly while working.
Freeing cambria gives u access to the hospital. But there is a SAM south.east of it (desert sam).
If u size cambria, got the SAM under control and found the chopper pilot u can make easy evacs for healing mercs there.
If u got no med pack, order one from bobby ray, since u freed drassen airport. (link on AIM website, select the correct destination for delivery!). Another option is let Ira train the guards for drassen, she is bad at aiming anyway and got the teaching skill which gives her a nice boost for training militia.

Maybe u should hire Fox. Shes sexy, has a nice attitude, got a medic pack with her and is a moderate doctor. Her dual hand skill makes her a nice fighter with dual pistol/mp5k/uzi and she learns quick during fights.

I notice u didnt created your IMP.
U should check your ingame E-Mail. The email for the psycho test isnt spam.. u can create your own merc which just costs u a one time fee of 3000$.
U should create him with good physical stats. If u drop skills like mechanic, medicine, explosives etc. to 0 u get another 15 points to assign but can not fulfill these tasks and can not learn it either.
Marksmanship is learned easy during battles so i usually go with a low marksmanship.. but if u create your IMP with marksmanship 75 or above u get a mp5k instead of the pistol.
Also consider hiring 1-2 medium priced mercs for one day at the start with a rifle and buy their equipment. Scope and Lynx are my candidates for that.