What was military medicine like in Ancient Rome?

Medicus - physician or combat medic. Specializations included surgery (medicus vulnerarius), ophthalmology (medicus ocularius) and even veterinary (medicus veterinarius). At least some held rank equivalent to a centurion.

Vesuvius preserved a fine selection of Roman medical instruments, including a tile cautery, bone forceps, scalpels, a catheter and a portable probe case.

LINK: http://exhibits.hsl.virginia.edu/romansurgical/

To quote I M McCulloch

Battlefield medicine - The Ancient World 2000 BC-AD 500

By the time of Trajan and Hadrian (AD 98-138) each legion of 10 cohorts (numbering approximately 6,000-6,500) had a legionary physician (medicus legionis) with medici cohortis assigned to each cohort of approximately 600 men. These military-establishment physicians were regarded as immunes, exempt from guard and combat duty and ranked as principales (non-commissioned officers). In the elite Praetorian and city cohorts, the physicians were required to be Roman citizens, while the physicians of the vigiles and auxiliary troops serving in Italy and the Roman provinces could be freedmen or foreigners. For this reason, the staff surgeons of the Roman army outside Rome were called medici ordinarii. Legionary physicians were considered all to be of equal rank, had no immediate medical superiors and were subordinate only to their camp commander (praefectus castrorum), or in his absence, the tribunes of the legion. It was their task to supervise the food, clothing, encampment and general hygiene of the troops and to run the camp hospital.

The greatest contribution of Rome to battlefield medicine was the establishment of the hospital system. Each military camp had its own travelling valetudinarium to accommodate the sick and the wounded. According to the writer Vegetius, hospital personnel consisted of hospital superintendents (optiones valetudinarii), physicians (medici castrorum), and sanitary personnel (capsarii) who carried dressing materials in a pouch and were attended by pupil understudies (discentes). A turn of the century excavation of a static Roman hospital at Köln, Germany revealed a well-laid out structure with design features that indicated awareness of infection and ways of preventing its spread. This hospital could accommodate up to 220 patients comfortably with 38 small wards, a dining room, two exercise quadrangles, sewers, water-piping, a heating plant and a kitchen.

I will leave your other questions for someone more qualified! I am a very amateur historian.