Lets keep one thing in mind as we troubleshoot your computer, the software isn't the problem here, the problem is selecting the correct input.
Spin It Again simply records whatever windows provides for it, so if you are feeding it bad sound, it will record bad sound.
You mention that you hear music AND whatever else is going on in the room at the time, this can only be caused by one thing...you are recording using the microphone in the laptop, or other device you have connected to the pc that has a microphone. Do not use that option.
Unplug anything you have plugged into the computer that is connected via USB and is an audio device of any kind, like web cams or microphones or audio interfaces.
Plug in the turn table, run SIA and select the USB option as the recording device. That's all there is to it.
If you get no sound at all, you need to look on the turn table manufacturers web site for an updated driver to install. Drivers make all the difference in the world, they are the translator between the hardware and the windows operating system.
To test the above statements, download and install any recording software and try recording the turn table with it, you'll have the exact same issues until you get the input selection correct and the driver for the turn table updated (if applicable).
Lastly, right click on your speaker icon in the system tray (by the clock) and choose "recording devices". Find the USB turn table and right click on it and make sure it's not disabled.
From that same place, you can choose properties, then select "listen to this device", this will allow it to play back through your computer speakers as it is being recorded. Making the connection directly to the stereo unnecessary.
Here's what mine looks like