Where do you place a villager in order for him/her to study medicine? I have tried everywhere around and inside of the completed hospital (and I have several villagers with healing experience), but all I get is, "Trying to fix the hospital."

You place them on the Herbs, not the hospital. The hospital is for sick villagers to gather to when they are sick.

I believe they will only study up to the adept level, as well. Mastery requires them to actually heal.

"Studying medicine" occurs in VV3, and you may have seen posts about this laying about.

It's interesting that until VV3, Healing to me was an ignored skill for me. If they became Masters at it, it was completely by accident. Since there wasn't a way to train them quickly without baby-sitting them in some way, I never used it for Esteem Elders (intentionally, anyway). Now in VV3 it's one of the easiest skills to master.

I prepare a special stew that makes the other villagers sick and then I use the trainee doctor (or children) to cure them, and then they improve in the skill. And I make all the villagers eat from the stew until is gone, and then the doctors go on their own to cure the people.
And I take the doctors to study all the flowers around the island. And just let them be... but if you start when they are children, by the age 30 they'll be master doctors.