Do different routers affect sound quality

Anything is possible I suppose but as long as a router is in good operating condition, I'd put that way way down on the list of things I would fiddle with in hope of better sound.  Probably so far down I would never actually care enough anymore to fiddle with it.

Get your router off of the wall wart or equivalent that feeds it power.

Power it from a linear power supply with a well constructed DC cable and connectors. 

Use a decent LAN cable. I use Supra CAT8 which is 'reasonably' priced. There are a number of reasonable options.

If you want to go a step further, add a LAN filter.

All of the above can be done for reasonable investment, as long as you are wise with your choices.

The switch is a good recommendation, but they suffer from the same issues which can be addressed in the same way as the router, but you are essentially duplicating the process.

If you can avoid the switch, you'll have less to address.

I have moved to sub networks and pull WiFi into the sub router that has a single LAN out port for my audio needs. This also allows me to keep my cable router far away from my audio and video systems.

TV feed is completely separated from audio with another sub router.

Another alternative is to go to a fiber based system.

It matters. Do not ignore the very, very front end of your streaming / network based audio system. For a minimal spend (relative to the cost of regular components, cabling, etc.) the results are outstanding.

What's a wifi sub router with a single lan port? Is that anything like a wireless range extender?

No, but the wall warts do. Keep them outside of your AC power conditioner.

Also, keep wifi devices physically far away to minimize chances of picking up RFI/EMI

Also, USB from Laptop is very good, but often the Laptop power supplies are crap. Same advice as above. Keep the power supplies away from your audio electronics.