

Am I correct in understanding that the card will run at PCIe gen 3 X1 if I do this?
Correct. The situation you described in the original post would result in Gen 3 x1 speeds.
The interface will always default to the fastest standard that both sides can support. If one is gen 2 and the other is gen 4, gen 2 is the highest that can be supported. If one side is x8 and the other is x4, x4 is the highest that can be supported.
What can I do if the card is PCIe gen 2 x8?
If you put a Gen 2 x8 card in a Gen 4 x1 slot, you will get a Gen 2 x1 link.
Yeah, going along these lines. There is probably a USB header on the motherboard. These have pretty darn good speeds. You can get an adapter that lets you turn those into a USB-C port and then use a standard USB-C to Ethernet adapter. Something like this or this. No guarantee on either of those specific adapters being good though. Looks like slim pickings for such things and both of those are garbage brands.
If you have a USB-C port on the back of your motherboard, you can get an adapter for that directly.
Also, motherboards generally come with 2.5Gb/s ports now too. Some even have two. Something to consider.