For your 1st question, if your ethernet cable is old and damaged or just has a lot of cross talk within it then the cable will affect the speed information can be passed along it so yes, a losy cable can affect things very badly.
For your 2nd question, generally when selcting ethernet cables, there are a number of choices but their use is limited to what you are connecting ethernet cables to.
If you was talking about connecting a PC to a router and you have other ethernet devices connected to the router via ethernet cables such as NAS drives or other PCs sharing files etc. between them, if the PCs, other devices and router support their speed then Cat-6a cables would be beneficial for each device to the router as they will offer faster file transfers.
For the use you are asking for (between the router and modem), a Cat-5e cable will suffice as the use you get out of that cable is Internet connection and not networking multiple devices. You probably don’t even have over 1 Gigabit Internet, so you won’t see any increase in your Internet speed if you use Cat-6 cables instead of Cat-5e.
See this site for more detail