Does Tor affect ping for online games? The servers to which I usually connect are in Europe. Is there a way around?
When running the browser bundle it only sends the data that goes in\out of the firefox portable browser that's included with the bundle through tor. You say you play BF4 and that ping is not affected - it should not affect BF4 at all - since that traffic is not routed through tor.
If you want to relay all traffic then you need transparent proxying or a 'middlebox' routing everything, which is not recommended on a windows OS machine, because of leaking.
Battlefield 4 is windows only, and also UDP communication is not supported through tor either, so playing games over tor like this makes no sense.
In regards of your youtube video question - it shows them with html5 video player instead of flash.
When you are using Tor, your traffic not only goes through the internet routers but among them passes at least 3 other routers (onion routers) these ones have to decrypt part of the information and forward it, which would cause additional time too, so the answer is yes.
Tor exit nodes are used by many users at the time and usually their bandwidth is slow.
But Tor works only over TCP protocol so UDP packets don't go over Tor and as far as I know, most online games for which the ping matters use UDP because TCP is not suitable for them. You may have heard about the lag (lost packets) that is because UDP.