In these "Single Phase" motors, you only need to use the cap for changing the phase, thus artificially creating a third phase, so no need for the resistor (it's the same thing with the motors on celling fans), so that's that. If that was intended to be a RC snubber it's in the wrong place, however you still need a cap for that third phase genration,
About the relays, these motors usually have a high inrush current, and relays (at least the ones I bought last time from finder for the same application) work fine for just connecting and disconnecting the power from the motor, but you don't mention how you want to change the rotation. In my experience, they work fine connecting and dis-connecting 150W ebm HVAC intended fans, so your 14W motor should be OK. (the rating of that relay is what makes it appropriate to your application). This is a standard way of controlling the power flow to fans in Building Mass Control (BMS) systems.
One alternative is using a Triac, driving it with an optocoupler (same way you control your relay with a transistor on your low voltage side). a Triac in this setup would also allow you to have some sort of an open loop control for the speed of your motor (same procedure used in dimmers and celling fans). you can buy the built module on ebay for like 4$. but if you design it yourself, then you would definitely need snubbers (or other methods like TVS diodes rated on high enough of a voltage) to suppress the spikes when disconnecting the motor but I remember working with a 800V triac on a 150W 230V motor and it worked just fine.
Also, zero crossing is used to make sure that you switch it at a time when the voltage is close to zero, reducing harmonic noise and the spike generated, but with proper supressing circuitry, it wouldn't be necessary.