If someone was severely injured in combat (such as loosing fingers and partial use of a hand), would they be discharged or reclassified?

If you get hurt and you can do other jobs and still complete Soldier tasks like shoot your weapon and wear body armor, the Army will reclass you to a job that fits your limitations.

If you cannot complete those Soldier tasks the Army will medically separate or medically retire you, based on the severity of the injury and limitations

They would be referred to a MEB where they would be found fit or unfit for duty according the the 40-501. At that point there may or may not be options to retain and switch to a different MOS with different physical standards.

In this hypothetical situation can the individual still do their job?

It's not unheard of for those who are injured redeploy and keep doing the job. Happens all the time. I had a LT who was injred on deployment and lost a foot. He deployed with the unit a year later to Iraq. Dan Crenshaw deployed twice after loosing his eye. Lreoy Petry lost his right hand and deployed again. It's not a deal breaker but you have to be able to do the job.