When I use a medkit in battle, does it 'consume' the medkit? I know I can't use it again that battle, but will that same medkit stick around and be available to use in future battles?
No. Medkits remain after use.
They're similar to the Support's Smoke Grenade or the Ghost armor's stealth ability, both of which have limited uses that recharge every mission.
For what it's worth, this is how medkits worked in X-COM as well.