I want to use 2 imacs (one with FCPX) in same house with Airport Extreme I have a new 2017 iMac, that when it arrives, I'll be using in a home office to edit videos for video production business on Final Cut Pro X (FCPX).  The 2017 iMac will be hooked up the same as I currently have a late 2013 iMac.

Cox coax cable to router to Ethernet cable to Airport Extreme to Ethernet cable to back of iMac.

I want to use the 2017 iMac solely for video editing and hope to keep it clean and fast, but would like to keep the 2013 iMac downstairs as a home computer.  What exactly would I need to make that happen. A second cable internet line and router for downstairs and home computer?  Seems expensive 

I don't want to lose any speed or power upstairs with FCPX and the new computer.  This is paramount.  

Would I be better upstairs in video production office to cable from Internet router straight to the back of iMac and not to Airport Extreme? 

I'm sharing a house with 2 teenage daughters and a wife with iPhones, MacBooks, etc.  I'm hoping someone will feel my pain and help out a newbie with this.
Your Internet connection has nothing to do with the performance of your hardware with respect to video editing.  The only place it will come into play is when you are transferring data files over your network.

If this is of concern, the answer is simple:  wired is always better.  With wired Ethernet, you're not sharing bandwidth like you would with WiFi.

That said, if you aren't transferring files over the network and all the work will be done locally, it won't matter if you are going WiFi or hard wire.

(IMO) I would hard wire both my iMacs to the network and use the WiFi for convenience like your daughters' MacBooks, iPhones and iPads.  It may cost a bit more during installation, but the reliability and stability will pay dividends in the future.