<< 1. if I throw this MP3 into a DAW, is there some way to remove the noise? >>

It's possible but that's a temporary fix and not the solution to your issue. It's likely to only reduce or mask the noise rather than remove it. The solution is to stop the noise at the source.

<< 2. do you think the existing sound card created the noise? >>

No. It could be in the sound source chain, Mic, Cable or sound card's physical connection. The symptoms detailed throughout this thread make me think the noise is externally generated and being picked up along the way into the PC.

<< 3. if I get an audio interface, can I render an MP3 or WAV without noise? >>

It's possible but you'll have to spend the money to find out. Since you know you are currently recording your audio in an external device not connected to your PC, it's a clear indication your PC system including the soundcard is how the noise is getting into your recordings.

It's possible the noise is being 'captured' by the soundcard but just as likely it's some other on-board system or component of your PC. No guarentee an audio interface will 'solve' the problem. It's also possible one brand of audio interface works and the noise is gone while if you choose another brand, it doesn't.

<< 4. this is totally counterintuitive to me, but could the rendering process be picking up the room fan? >>
Yes. It could be your house wiring, plug, a light switch, a power strip, florescent lighting, led lighting, AC, fridge, stove/range, power supplies or a host of other items. It could also be coming from the PC itself...

You can test the audio interface theory with your Zoom H1. Download and install the drivers, connect the USB and set the device as your recording and playback selection, make a test recording and you'll have your answer.

BIAB Ultra Pak+ 2024:RB 2024, Latest builds: Dell Optiplex 7040 Desktop; Windows-10-64 bit, Intel Core i7-6700 3.4GHz CPU and 16 GB Ram Memory.