1) by settiing MedKit heavier you basically created the dedicated Medic already. Lets use weight management to let player create a role. I would not recomend to add skills like : medic, sniper, runner. Adding more skills means soldiers are less versatile AND player have to keep attention to more numbers. I would rather see Xenonauts 2 be a number low system but interesting choices&consequences.
2) yes, the emergency status ( bleeding, poisoning, burning, acid, shock, ... ) removal should be one action. Then healing HP should be another one. However, for sake of simplicity lets make it so, that Medic always remove emergency status first. It is also logical, that medic takes care about critical status first like artery bleeding , then takes care about blister on your little finger.
Regarding healing HP it should rather be "healing over time" than an instant spell. For sake of reality feeling, apply HP heal, then each round soldier gets 5HP until healing effect is depleted.
3) Althou it sounds reasonable it would require additional dialogue in the combat UI. Combat is turn-based and slow already, adding such level of control would make it overly complicated. So not recommend. Simple UI is key to success. Check gamee : Trader 40K by Owlcat games. They made so precise and overcomplicated battle UI that is unbearable.
4) Medic should be able to resolve various emergency statuses AND shock statuses (low blood pressure, shell shock, hazard inhalation, unconscious, ...) these shock statuses means, target is TU limited or even zero. So Medic emergency action can only consume medics own TU. However, medic healing HP action could require coordination. So target will consume same TU as medic. Which means, target can not shoot while he is receiving HP healing, that makes sense.
5) yes, Medkit should have a capacity. Similar to shield mechanic. So Medkit 82/120 HP capacity means it has 82 points of 120 maximum. Also, weight management is important - each point of healing capacity has certain weight. So Medkit 60/120HP has half of the original weight.