Wells don't actually have to be designated as a water source or a meeting place. Dwarves will preferentially take water from a well any time it is needed. They'll only use a backup source if all the tiles next to the well are occupied already.
In terms of "why isn't my dwarf being treated", take it one step at a time.
Step 1: is the dwarf Resting? As in literally performing a job called "Rest". If the dwarf isn't Resting, no medical treatment will be offered. Ever. If the dwarf is in the military, you may find it helpful to remove him/her from the military for the time being. That has sometimes worked for me to switch a dwarf from "no really I'm fine it's just a flesh wound" to "Rest".
Step 2: has the dwarf been taken to a bed? The bed doesn't even have to be in a hospital. If there are no hospital beds, they'll take the wounded dwarf to his room, or to a dormitory, or whatever.
Step 3: (v)iew the dwarf and see what the (h)ealth screen says. That screen has 4 or 5 tabs, so be sure to check all of them (the critical information is on the second to last tab, I think). The critical one says what needs to be done -- "Diagnosis required", "Crutch required", "Needs suturing", etc. The last one says what has already been done, by whom, on what date.
Step 4: are your medical dwarves idle? Health care jobs seem to have extremely low priority, especially if your medical dwarves have any of the personality traits that don't match up well (hates helping others, procrastinates).