Yes you can record 2 mono channels. Just choose only the L or R input in Reaper for your tracks.
To use your Mixer FX without recording them you would need to come out of the insert points using them as a direct out. you can't just plug a jack in there for this though...This article explains the options clearly.
http://silentsky.net/wordpress/archives/113
So channels 1 and 2 of the mixer inserts connect to the UCA as described in the link and the UCA outs connect to ch 7/8 phono inputs.
So you have DAW playback level adjustment on ch 7/8, mic or line ins on 1 and 2 and can add FX to any channels that will be playback only.
There is one possible issue here and that is the direct outs of the mixer may be a little hot for the consumer level UCA inputs.
I suspect you'll be fine if you don't overdo the gain but maybe try to test one of these connections with your mic first.