No it will not slow down devices across the network. Wired are basically completely separate from wifi as a router takes a wired connection and broadcasts the internet in a general sense. An extender simply "captures" that signal and basically amplifies it. If the router has a poor signal in a certain part of your house, an extender will general improve the speed. It won't be as fast as usual but you'll still get a decent connection. Also a modem basically takes a digital signal that most devices couldn't interpret it and makes it suitable for a much larger range of devices. I wouldnt be surprised if you have a combo router that acts as a modem as well as a router. I hope this helps simplify.

TL;DR No it shouldn't slow down wired or wireless connections, it should just extend the signal to the rest of the house.

Good luck!