You said you were using a cheap mixer? So your mixer output goes to the sound card input. These are line level. The mixer handles the inputs - the sound card just records what comes out of the mixer, so has no part to play in selecting vocals or lines or instruments.
The BIG problem is that you are using the mixer outputs for your monitor speakers, I suspect. In this case, many people would use the mixer aux sends so that what you are listening to does not have to be the thing you record.
Routing is always a fudge - but workable. You simply use the PRE-FADE send on the source you wish to record. This will be mono - so the option in your picture to use just the left or right input is the correct one. If you wish to record something in stereo - like maybe a keyboard, this will only work if you have two pre-fade sends. Do you have two? The sound card is so simple that all your decisions get made on the other equipment and software. Cubase, for example is happy to have two inputs - so when you create a new track as a stereo one - it would take two input channels and record them, but if you select mono - it records just one of them.
On playback, you can use the multiple outs from the card if you wish - or not. So if the singer needs to hear a track through headphones - you could feed that out of say output 3, and stick that back into the mixer for them to hear. All these things really don't need you to do anything with the card.
The only downside of this card is that even when new, the latency wasn't that good. This probably hasn't got any better.