Yes, that's about what I was thinking too.
You can look in the benchmark thread to get a more accurate estimate for TF2. Take your current fps in 6s/pubs with the settings you want, run the demo and then based on the ratio you should be able to get a rough idea of you need by comparing the demo result with results in the thread.
However I believe that you'll probably have to overclock. Since you're already going for the 8350K and need a cooler anyway the only difference would be a Z370 mobo. That would also allow you to run faster RAM (TF2 loves fast RAM, especially timings are important) since B360 is limited to 2666. 3200 CL16 shouldn't cost much more, there's even one 3600 CL19 kit in stock for 167€ right now.
There are cheaper cases with two fans, the for example Focus G should work, I think, but there's probably other options as well.
The way I see it you've got 3 options:
1. Optimize for TF2. Z370/Z390 mobo and faster RAM, GTX 1060.
2. Optimize for the other games. Usually I'd recommend the 8300 so the drop isn't too much but you save 30€ and don't need a cooler but that shop doesn't even offer it and the 8100 is almost twice the price it should be so I'm not sure it's even worth it. Simply getting a cheaper case would get you back within budget so it's better to do just that.
3. Go over budget and do both. With the RAM I mentioned, the Focus G, cheaping out on the mobo (will limit your overclock a bit) and getting a 250GB SSD instead of 500B you can get it down to 1230€, which isn't too bad I guess.
Ideally you'd want to keep the 500 GB SSD (+30€) and get a mobo with decent VRMs, but that would cost you at least another 50€ in that shop.