The "put the plug half way in" method CAN be used but it's a bodge to get around the limitations of an unsuitable mixer.
What you want in a mixer will depend on how you record. If you want to record multiple tracks at the same time, then you want a mixer that offers "direct outs" on each channel. If you only record one or two tracks at a time, then a simpler mixer will do--just use the main left and right outputs.
For your monitoring question, the best way to do this is to use a mixer that has at least one "Pre Fade Auxilliary Output". All but the cheapest, most basic mixers will have this. To use it, bring the line output(s) of you sound card back into the mixer. Turn up the appropriate Aux controls on each channel you're using to give you a mix of the DAW line out and also whatever new instrument or vocal you're recording. Being a pre fade aux, leave the actual faders for the line in completely down but turn up the faders for the new material to feed this into the computer. You can set the exactly balance between new and old tracks by adjusting the Aux send knobs on each individual channels.
Notes: Most mixers allow you to PFL the Aux output to feed it to your headphones...this should be something to watch for when choosing your new mixer. However, if you don't have this feature or need to feed multiple headphones, use the Line Out for the Aux and feed this to a headphone amp.
Second, to avoid possible latency issues, it's desirable to be using ASIO drivers for your sound card/interface.
Hope this helps,
Bob