There are a couple of lan testing software. You would need 2 devices inside your house. IPERF is a very old line mode command that is used to test just the network part of a pc but it will test your cables at the same time. You pretty much plug 2 machine into the router and see how fast you can get. In most cases you will get very close to the port speed either 100m or 1g.
Generally when people suspect they have cable problems they just try another cable because it is so inexpensive and take less time. There is almost no difference in price between cat5e and cat6 so buy what ever is cheapest. The main difference in a patch cable is going to be things like the flexibility, what the outside jacket is made of, the type of ends and of course the all important color. But since they all run only as fast as the port allows them to you will not see any performance difference.
Once you are very sure your equipment in your house is fine then you need to call the ISP there could be problems with the connection to your house.
With FIOS if verison has the ethernet port on the ONT active plugging directly into that with a pc tends to be your best way to eliminate equipment in your house as the cause of the problem.