Well since sports is what I shoot the most of let me toss my 2 cents in here.
First yes the 80-200 is useable for both Baseball and Soccer within the limitations of the lens. Baseball, infield and short outfield, soccer, close action. You will have to plan to shoot to the limitations of your gear.
As for a TC. Don't bother, you loose IQ and in a lot of situations that DOF you are wanting to get a sports shot not just a sports snapshot. In general you want as shallow a DOF as possible to capture the action only while the foreground, back ground and all other players not on the same plain as the action are OOF. That is one of the key components of making action pop.
You need a minimum of two bodies, three would be nice for pre and post game as well as halftime/between innings etc. You generally do not have time to fumble with lenses. When on the sidelines for football or soccer I generally have my main body with a 400mm or 300mm f2.8 attached. A second body has a 70-200 f2.8 and the third body has a 24-70 f2.8 with a flash unit either attached or at least handy for the pre/post/halftime stuff.
You need fast glass. You need long and short glass. My sports lenses range from 11mm to 400mm, including a 200 mm f2.0, 300mm f2.8 and 400mm f2.8. None of my glass is slower that f2.8.
Do I have a Canon 1.4 and 2X TC? Yes. Do I use them? Yes. Do I use them for sports? No. I use them for nature/wildlife on the 300mm and 400mm. The IQ is fine with those lenses when shooting wildlife and since the wildlife I like to shoot has the potential to Kill you, (Bear, Mountain lion, as well as Deer, Elk, Eagles, Hawks and other critters) I want my distance.
My recommendation to you for now. Learn to use what you have. The 80-200 is a good start. Learn the assets and the weaknesses and plan to shoot to the strengths. Will you miss some stuff because of the limited range? Yes. Frankly some of the most spectacular stuff I have shot has been with my 70-200 since the action was coming my way, close up and personal.
Final recommendation, if you are serious about sports shooting, start saving money now, get a part time job or whatever, make a plan of what you need and in what order in terms of glass and bodies and start saving. Buy what you can, when you can, as you need it. What I mean about that is, if you save up enough for that 400mm f2.8 but it is now December, well, you might want to take some of that money and invest it in a nice 35mm f1.4 for Basketball, keep saving and get the 400mm in the spring when you can use it for Track and Field.