It's looking better.
Yes, weapon handling will let you use the weapons.
http://www.fo2238.fodev.net/wiki/First_aid As in the table, 75% heal will have a 4 min timeout for first aid so that should be good enough. You'd be able to maximally heal 72 hit points with that much skill.
You don't have a lot of hit points though. You can die in one critical shot or a burst. Maybe instead of taking "even tougher" and "sharpshooter" you'd take two levels of "lifegiver" which would add 50 more hit points giving you 189, which is better. Still, some shots can get you for 200 hit points. If you'd take all three "lifegiver" perks you'd have 229 hit points
If you would want all three "lifegiver" perks, I recommend you take them instead of "even tougher" "sharpshooter" "action boy" (you have "bonus rate of fire" anyway and if you're in real time mode it'll be just a second before you can fire your second shot [or you can always add more agility to make up for it])