When was I first supposed to have learned the name on the Health Center
Issue 113. No one would be on duty because it's not an emergency room, it's a campus health centre. Taking Amy to Emergency would mean all sorts of unwanted inquiries - she was shot, after all, and presumably there's some pretty overwhelming interests on everyone's part to keep this whole thing hush-hush and make sure to keep the police out of things.
why didn't Adrian take them to Marli?
I'm guessing because, as a result of total luck, Amy was only slightly hurt by a bullet grazing her, they had access to the Bendo Health Centre, and Marli is already busy with stuff when they can take care of things themselves.
As such, why would the health center be completely deserted?
Middle of the night? It's still a small campus in a small town, after all, and as employees both Amy and Hayley wouldn't necessarily attract suspicion, especially if one of them needed a bit of (relatively) minor first aid.
Just my take, of course. The usual disclaimers (i.e., that I don't know anything about what's actually going on) apply.