New Home Networking Setup

Hello Everyone,

I would like to start off by apologizing if my question has been asked on this forum before ... I've been searching for an answer for over 2 weeks now ... and finally decided to post the question.

New Home Network Setup

Basic Information;
House Size: 5000 SqFt
Floors: 2 + basement
Main Router: Installed in basement, with 3 direct wire connections and has wireless capabilities
Connections: 1 x in office on main floor + 1 x in family room on main floor + 1 x in the hall way ceiling on 2nd floor

I'm planning on buying an Engenius Router (most likely the EAP300) to hard wire on the 2nd floor ceiling.

For the other 2 connections (on the main floor office and family room) I would like to purchase minimum 1 wireless router and the other one would only be a router (or maybe its called a "switch") without wireless capabilities.

Query # 1
I want to have a wireless router on each floor (basement already has the main router with wireless capabilities --- second floor will have the Engenius router) and I would like to set them with the the same SSID (name) . Basically since I will have three wireless routers on each floor ... I want them to have the same name ... so that regardless of whichever floor I'm on .. my devices get the best signal.

Example: If i'm on the second floor .. I want my device to automatically switch to the 2nd floor router so that I get the best signal .. and now if I go in the basement I would want my device to automatically get the strongest signal from the basement router ... so because I will have three wireless routers ... I should have the best signal everywhere I go ... and I want to name all the wireless networks the same name .. but I would also want them to speak to each other and pass the best signal ,, hopefully I explained that right ...

Query # 2
What wireless router do you recommend for either the 2nd floor and for the main floor ... currently I have not purchased anything ... the only 1 wireless I have is in the basement .. provided by my service provides.

Thanks for reading ... any help is greatly appreciated.