I would use the 70-200 or 70-300 for portrait work and sports. Portrait work might on occasion be for pay...sports for my own personal interest/family. Sports would include cross country (could use either lens), baseball and basketball (might prefer 70-200), and indoor cheer competitions (definitely 70-200).
I also like to shoot random things, animals, landscape and scenery. I used the 24-105 almost exclusively on the 5dmk3. I often found myself wanting more reach. I thought the 24 was just on the edge of not being wide enough on FX at times. I had the Canon 70-300 at one point and hated it. I did not find it to be appropriately sharp. Bad copy? I'm not sure. I liked the range but not the lens. I used to have the Canon T3i with kit lens and this is where I used the 70-300 primarily. Also had the Sigma 10-20.
My goal is to have the best image quality available (I really could care less who makes it, Canon, Nikon, Sony...) in the most comfortable/lighter weight system. The 5dmk3 produces beautiful images. The only negatives are pulling shadows too much can get ugly, and the weight/comfort factor. That camera just does not fit comfortably in my hand. It feels awkward and I always feel like I'm going to drop it. Yes, I use a strap. I'm just saying, if I grab it real quick it just doesn't make a good fit for my hands.
So, since I do sometimes shoot from ground level I like the tilt out screen on the D5200. I basically ruled out the latest Canon crop cameras because the Nikon's have far more to offer at any given price point. There are some great deals through later this month on Nikon body and lens combining too. I'm thinking I will go with the 70-200 f4 but I am still stuck on the 18-105 kit lens, or 24-120 f4. I know I will want wider than 18 on DX so I'm just not sure. Maybe the Sigma 10-20? I was pleased enough with that lens in the past. I just want the best wide, mid and tele for me and I guess only I really can know what that means. I appreciate all the information. I'll probably finalize by mid-week and order something up.