Can you control a 12V Solenoid Valve using motor dev kit MCLV-2?
Yes, you probably can.

If you happen to have this board around and want to test, fine, although its not geared toward driving solenoids.  The IR2181 is not a processor, its a mosfet driver.  The high side drive cannot drive 100% duty cycle, perhaps 75%, depending on bootstrap cap and configuration.  This means you can't just turn on the high side and leave it on.  So you would probably connect to motor pins depending on duty cycle wanted.  If 12v and 100%, you couldn't use the high side driver, but rather high side of supply and low side of driver connected to solenoid.

I don't see why you couldn't drive a 12v solenoid at 50% duty cycle with 24v supply.  Don't forget that you can supply the board with 12v if you want.  The MC datasheet dc output rating totally depends on the users supply, perhaps you misunderstood that one.  It all works fine at 12v.  However, if you use 12v, you have to factor the high side max duty cycle.