The UMC22 is probably not the best choice of interface, given its limited inputs.
The UMC202 would have been a better choice.
But your basic set up is sound. You can hook up drum machine, mike and guitar to mixer, then take one of the main line outs into the UMC22, and from there by USN into the Lenova.
For your playback, you will need to connect speakers or headphones to the UMC22.
You will need to go ito Cool Edit and set its recording and playback device to be the UMC22. Likewise for the Lenova's general sound settings. Note that CE 2.1 is pretty old, and you will probably find newer DAWs to be more durable.
With this set up, you are face with some limitations:
1 You can only record one track at a time (unless you use the work around suggested by Keith ^^), which means your drums will be just mono.
2 Or, you can record drums, vocals and drums all at the same time, but they will all appear on the one mono track.