Welcome to the forums. You're talking to other customers here.
ONT's are provisioned by Verizon to have the internet connection (WAN) active over coax or Ethernet, but not both. To change from one to the other you must contact Verizon.
You can use your own router with FiOS, but know that Verizon will provide no (zero, nada, zilch) support for anything other than their router. Further, FiOS set-top-boxes do not work with a non-Verizon router. If you have FiOS TV, you must have a Verizon router in the network, and it must connect to coax (although the WAN connection can be Ethernet).
Depending on the construction of your home, running a new coax may not be that difficult. While you can make your own cables, it is often easier to purchase a pre-terminated cable. You can get a 50-foot RG6 terminated cable for not that much money and you won't have to purchase any crimp tools and extra connectors (in case you mess up). This work can also be done by any competent contractor, it doesn't have to be Verizon. A "Home Theater" installer may be the most cost effective.
If you do make your own cables use RG6 and compression connectors. The one you highlighted looks decent, although you might want to check other online stores for better pricing.
YOU need to use a verizon router somewhere in the mix. The ethernet connection can be used but both the coax to the TV's mus also be connected the Verizon router. That's because the TV's do need to talk to the router to get to the internet for several functions. I.e the router must have Both coax and ethernet connections to the ONT. And Verizon needs to enable the ethernet connection.