It doesn’t matter what cable you use on the inside of your house ( LAN) You will not change the speed of your Internet (ISP WAN) unless they change it. That being said, what you will change by upgrading the cables is the speed at which 2 or more of you cabled devices ( NAS, computer, servers etc.) communicate. As far as the Ethernet cable is concerned, I wouldn’t spend money on anything but CAT 6 maybe CAT6A. The rest of them are not worth it. If you want a better cable for really short distances use DAC cable SFP+

Some things sound off to me. 1 The ISP wouldn’t send someone out since the previous tenant owes them money. That means that it had internet there before. This means you do have the connection from the ISP to your house. 2 Not all the ISPs have modem/routers that you have to plug into an Ethernet cable. Some have the COAX IN and Ethernet out. 3 where is the wire from your wall RJ45 terminated? Meaning - where does it go? Do you have a network closet that you haven’t seen yet? Some pictures would really help troubleshoot. Check your attic, closets, garage for a “ box” in the wall with some Ethernet cables, coax cables.