Just to preface with a great resource that might be accessible to those on the east coast of the U.S.: there's a great Civil War medicine museum in Frederick, Maryland that covers much of what I'm about to touch on. If you are even remotely interested in ACW (American Civil War) medicine, check it out.

To answer the last portion first: yes, doctors of the 1850s and 60s did attend medical school and had something similar to a residency period as we know it today. A person who wanted to be a doctor on the level of a commissioned army doctor (but especially a surgeon) had to go to school, and would have needed to have some level of experience. Of course, in the north as well as south, large battles sometimes saw the temporary enlistment of "doctors" who wouldn't have otherwise been considered for full-time duty, but speaking in generalized terms: doctors of the period were trained professionals, yes. The restrictions for being a nurse were much looser in terms of training (most anyone could volunteer, though the scope of what they were doing would have been limited if they had little or no training), but doctors were expected to be medical school graduates.

As far as medication, there's a myth that the ACW witnessed surgery (amputations, largely) without anesthetic on a massive scale, but this is simply not true. In both the north and the south (but especially the north), surgery WITH some form pain killer or anesthetic was the norm, not the exception. Where available, ether was used to sedate patients for surgery, and opium derivatives were used for pain management. Again, though: this myth of surgeries being performed with nothing more than a bottle of whiskey and a thick stick clenched between one's teeth was just that: a myth. By and large, medicine was available and used extensively by doctors on both sides during surgical procedures.

Of course, penicillin was a ways off in terms of development, so treatment for infections was difficult (so the fevers OP spoke of), and germ theory was in its infancy. Although many doctors were starting to come around to the idea that washing one's hands/instruments decreased incidents of infection (etc., etc.), reports from the period speak to surgeons clenching a knife between their teeth, and wiping the blade on a soiled apron between surgeries. They were a bit smarter as it concerned their water sources, but as you probably know, dysentery was a very, very, very serious problem for the armies on both sides.

So to answer your question, no, the doctors of the period weren't as knowledgeable as today's physicians, but this is entirely understandable considering the 150+ years of medical developments that separated them from us. The doctors of the period were hardly stumbling around in the dark, however, and did have a solid grasp on some of the basics of germ theory, and best practices as it concerned water consumption and treatment. Most importantly, however, this myth that civil war medicine consisted of booze, hacksaws, and a lot of screaming isn't rooted in any kind of reality. Anesthetics and pain relievers were available, and used extensively.

[Sources: Bruce Catton, 'Grant Moves South' & 'Glory Road'; James McPherson, 'Battle Cry of Freedom']