As to whether you or a Pro should refinish the floor, I don't think it's a question I/we could easily decide. There are a lot of factors that go into that decision.
Among them is your level of confidence. This is measured in a variety of ways. Like, your level of experience, your 'ability' to take on a 'new task', your familiarity with the equipment involved, and so on. You're doing the right thing in exploring first what the job entails. At least after reading that, you'll have a better idea if you or a Pro should do the job.
RE: The 100 year old floor - If they're plain, old wood planks, and they're NOT like a traditional hardwood floor, I think you ought to opt out to have a Pro do the job. If the floor is a traditional floor, then go back and make the call after doing the research. Again, it's hard to say from here.
As to why the center of the room has no stain on it, I can't say why. I'm sure after refinishing either type of floor, you'll have 'uniformity'. This, of course, hinges on the question: Can the floor BE refinished (again)?
See How To Refinish A Hardwood Floor for starters. Determine the TYPE of hardwood floor you have (based on the refinishing process), and go from there. You can have a Pro give you an estimate along with having him give you some info about the floor, itself.