I know it has been asked before, but I'm still not sure if I understand the focal length issues of EF-lenses on APSC cameras in Canon system. I already have the EF-S 55-250 IS STM and I'd like to buy a longer telephoto - if I get the EF 70-200 will it give me effectively more range than EF-S 55-250 (200 x 1.6 = 320 mm)?
No, a 70-200mm lens will give you less range than a 55-250mm lens on both ends of the focal lengths. In order to understand the comparative fields of view yielded by an EF-S and an EF lens on an APS-C camera body in terms of the same field of view on a full frame/35mm camera, you must multiply the focal length of both lenses by the 1.6X crop factor.
Your EF-S 55-250mm f/4-5.6 IS lens gives an FoV equivalent to an 88-400mm lens on a FF camera.
An EF 70-200mm f/4 (or any other 70-200mm lens) mounted on your Canon APS-C camera gives an FoV equivalent to a 112-320mm lens mounted on a FF camera. What the 70-200mm "L" Series lenses give you is significantly better image quality and wider, constant apertures. The faster speed of a wide aperture telephoto lens is vital when shooting action or sports under challenging lighting.
A 70-300mm telephoto lens mounted on your Canon APS-C camera would give you an equivalent FoV of a 112-480mm lens mounted on a FF camera. But most 70-300mm lenses are not much, if any, better optically than the 55-250mm lens you already have. They also usually have the same f/4-5.6 slower variable aperture.
For more, please see Why do Full Frame lenses and crop body lenses exhibit the same crop factor when used on a crop body camera?
For more on the optical differences between the EF 55-250mm f/4-5.6 IS II, the EF 70-300mm f/4-5.6 IS, and the Canon 70-200mm "L" series lenses,please see this answer to What improvements I could expect upgrading from Canon EF 55-250mm IS to a Canon 70 - 300mm? (Hint: Not a whole lot)
A full-frame lens of any focal length (let's say 200mm) and an APS-C lens of the same focal length (i.e. 200mm) are the same (in this context). The difference is only in the size of the 'image circle' that the lens projects; the full-frame lens projects a larger image circle to cover the larger size of the full-frame sensor.
So in your comparison, there are no complications like having to multiply one lens by some crop factor and not the other. A 200mm lens is a 200mm lens. A 55-250mm lens has a wider/longer range than a 70-200mm lens. The entire range of the 70-200mm lens falls within a portion of the range of the 55-250mm lens.
No, it'll give you a smaller range. The numbers quoted on any system camera lens are the actual focal length, not the equivalent.
You can get a longer range with a 70-300mm fairly easily, but the difference won't be huge - you'd barely notice it in reality. More than that is available but expensive and heavy.
I'd suggest you'll do best by looking at the subjects you're interested in shooting, then look at examples of that sort of photography on Flickr and review their EXIF to see how they've been taken. Better fieldcraft and practice, hard work as that can be, is likely to be your best answer.