What type of noise is this?

[Cherbette]

And how might I effectively remove it without too much damage? I have these horizontal, scrolling lines in a few different recordings and cannot figure out how to rid of them.

[Sharc]

It's regular interlaced video. You see these scanlines and comb artefacts when you watch the video on a progressive monitor (PC) without deinterlacing it.
So you should just enable the deinterlacer of your SW player. It's a setting in your player (probably called deinterlacing or bobbing)
When you watch this video on your TV, your TV should do the deinterlacing automatically and you should get a perfect picture.

If you still see some lines you would have to apply some extra filtering using Avisynth filters for example, something like

QTGMC()
QTGMC(InputType=2)
vinverse()

[Cherbette]

It's definitely not interlace lines I'm seeing. The lines are scrolling, moving. They are there even after QTGMC

[Cherbette]

[Attachment 55576 - Click to enlarge]My result after the above script....

[jagabo]

Hum bars. Defreq() can reduce those types of lines.

[Sharc]

Yep, I missed it. It's some sort of TV channel interference maybe, or interference through the cabling. Requires some more processing......

[Cherbette]

Hum bars. Defreq() can reduce those types of lines.

I've fiddled with this one a bit but cannot seem to find a spot to place the box on the spectrum that even minimizes it

[Sharc]

Then you could try

AssumeTFF()
Cnr2()
QTGMC()
QTGMC(InputType=2)
dfttest(sigma=20)      #adjust the sigma

[jagabo]

Hum bars. Defreq() can reduce those types of lines.

I've fiddled with this one a bit but cannot seem to find a spot to place the box on the spectrum that even minimizes it

This will remove a lot of it:

SeparateFields()
DeFreq(fx=0.0, fy=57.5, sharp=19.2)
Weave()

You can probably use it again (maybe multiple times) with a slightly different frequency to remove more.

[Cherbette]

I appreciate your help jagabo. I will continue to test this method and see if I can get a somewhat clean result