If I understand your questions correctly, then:
1) No, if you already have a splitter in place connecting to an unused port on that splitter should not cause any reduction in signal strength.
2) What I would suggest is you use a MoCA adapter to provide hard wired connectivity in the basement. Look at an ECB 2500 to provide what you're looking for.
You may need to add another port to in additonal to the splitter that's already in place in the basement to feed the input to the MoCA adapter. You could change the existing splitter one that has an additional port, or add another 1-to-2 splitter after the one remaining port that's already availble.
The ECB 2500 would get it's feed from the coax splitter and provide an Ethernet output for hardwire connections in the basement. With that configuration you wouldn't need another router.