The TLV3501 is a fast comparator, and well-suited for your 10 MHz application. I'm using this part on the front-end of my Time Interval Counter design, for frequencies under 1 Hz and up to the 200 MHz area. It has a small amount of built-in hysteresis which should provide adequate noise immunity when given a clean input. You wouldn't want to drive a 50 Ohm cable directly with the output of the '3501, but it's fine for short traces and moderate impedances (I drive a FPGA with it).
Your ringing is most likely due to your breadboard layout and bypassing. What are you driving with the comparator output?
As for biasing the inputs. I have a 1/2 supply voltage divider, bypassed, with 1 Meg resistors from the divider to each of the comparator inputs. One input is then bypassed to ground, and the other capacitively coupled to the input protection network. I am trying to provide a one Meg input impedance, with switchable 50 Ohm option, but if you are only planning on 50 Ohm inputs you should reduce the biasing resistors to 1K or so. This will reduce the input offset-current-induced offset voltage. Not critical though when you have a strong input signal.
If your input comes from off-board, then you need a protection circuit. I use two series resistors, and a TLV diode to ground at the junction. For 10 MHz, you can probably use 1K series resistors, which should give good protection. I use the T.I. TPD1E04U04 TLV diode.