I'm unable to put the 4070 in the 1080's spot since I can't remove the 1080, as for some reason the clip won't release. I can put in and pull out the 4070 without issue though, the clip secures and releases just fine. So, w/e. I'll just use both cards. Why not?

Sounds messy and you'll run into pcie lane allocation issues as well as some games using the wrong gpu. Not very power efficient either.

Except I threw out all the extra cables I had for my PSU, meaning I don't have a cable to connect the 4070, since it needs 2 8pins. I can use the cable from the 1080 and it works, but then the 1080 doesn't have power and I can't use my PC since my monitors are expecting the 1080. Not sure if I can just switch which it looks for?

Getting video out of the 1080 while rendering on the 4070 is possible but why not just plug the displays into the 4070? The 1080 likely won't turn on without the extra power.

Anyway, I bought this cable: https://www.amazon.com/dp/B099X48CQN?ref=ppx_yo2ov_dt_b_product_details&th=1

Seems like a terrible idea considering how much power modern gpus are pulling.

and found out it doesn't work. Powers neither the 1080 nor 4070. Mobo beeps twice then nothing. Not a power issue, since I tried using only that cable and unhooking the old cable completely. Then I noticed that oh hey, this cable only have 7 pins in the PSU side. My old cable that works has all 8 pins. My old cable also has red ends on it, don't know if that matters.

Assuming you've plugged that cable into your modular power supply then you might've damaged either the gpus or the psu. Depends if the designers took cases like this into consideration.

That leads me to the title of this post. Finding the proper cable. Will just any PSU to PCI-E cable with 8 pins work? I'm not sure the exact model of my PSU since the label isn't visible without removing it completely and that's a mess I don't want to get into for now. The back looks like this picture though: https://www.gamersnexus.net/images/media/2016/features/psu-guide/rosewill-photon-750.jpg

Rosewill Photon 750W or 850w seems likely with what i can assume about your specs. Full specs list would be nice.

I would strongly suggest removing the old gpu instead of trying to mess around with a double gpu setup. Especially if your use case will have no benefit from it.