Network Setup Question (Mainly about Routers)

[LTD85]

Hi all,

I intend to keep  this short an sweet, so please just hear me out.  Im in my parent's house (still 😞 ) but I maintain their electronic gear.  After a remodel a few years ago in which they added alot of tile and brick, but no ethernet ports, the WiFi signal in the house became very weak.  I have been trying to get around the problem with ever more powerful routers, but during a chat with my uncle (who is a network admin), he told me to put another router in the basement of my house.  Please see the cute drawing below.

My House Overview (Blue Box = Wifi device)

The ONT is in the basement, while the only ethernet port is in the center of the house on the second floor.  I disabled the verizon fios router's WiFi, and have an ASUS RT-AC3100 which I use for WiFi connectivity.

Here is my question, which is what my uncle suggested.  Can I plug a wifi enabled router directly into the ONT, and then have an ethernet "out" port used to connect to the verizon router on the second floor?  The reason I am unsure of trying this is due to the verizon router's use of a MoCa connection via coax, for the house's TV.

Current Setup:  ONT ==ethernet==> Verizon Router ==ethernet==> ASUS Router

Proposed: ONT => Wifi Router A => Verizon Router => ASUS Router

Also, will this mess up IP addresses?  I have the verizon router handle DHCP.  What happens if I want that to remain the same?

[CRobGauth]

No.

You can only have Internet provided via ethernet or coax.

Not both.

[TomH121]

"the WiFi signal in the house became very weak. "

Under the present cofiguration can you move the router and wireless router 

easily to the middle floor ?

Great pix by the way !

Sure better than a thousand words !