It sounds like you're having two issues. One with tearing, and another with skipping.
The tearing can only (as far as I'm aware) be fixed by using vertical sync.. Unfortunately, the graphics card is in charge of that and its drivers are in charge of that card so they could easily be ignoring SFML's request for v-sync.
The skipping does sound like it's skipping frames; it's been mentioned on the forum a few times already and I've seen it on my own projects. You should make sure you've tried a release build instead of a debug build first as that often makes a difference.
You also mentioned that it could be skipping pixels because of using floats. I don't think floats would directly be the cause. It would only be if the two draw positions are two pixels apart. If you're rounding and the two positions are 2.49 and 3.5 (distance of about 1), the rounded positions would be 2 and 4 (distance of 2).
Comparing with League Of Legends is an odd thing to do. It's rendering in 3D for a start