“Will the ER do an MRI for back pain?”

Unlikely. With the exception of a stroke, MRI's need to be scheduled ahead of time. Most back pain is musculoskeletal in nature and can be treated with analgesics, muscle relaxants and possible physical therapy. If you have pain radiating down a leg, have weakness or numbness in a leg, then you need to be seen right away.

Unlikely as a general answer. Not all hospitals staff their MRIs 24hrs/day; when staffed, they have a schedule of inpatient and outpatient exams to do. An acute MRI is rarely needed for "back pain," unless in the setting of fevers; IV drug-of-abuse or immune-compromising or blood-thinning medication; or an abnormal neurological exam. Usually pain/spasm relief is sought, and if obtained, the patient is discharged home, to follow up with their Doctor or an Orthopedist for further evaluation and management as an outpatient. If it is a very urgent question, it is more likely that a CT scan will be obtained of the area of the spine where the pain is localized to, as CT scan is quick, usually staffed 24hrs/day but definitely more than MRI, and ER has ready access to it. 

Typically not unless there are neurologic or infectious concerns.

Unless you have objective physical findings, very unlikely