I see this has 4 ports on it. Do they operate as a switch or a hub?

Gigabit hubs don't exist so I'm reasonably sure it's a switch.

I have an Asus AC3100. I am trying to get a better signal in a room about 50 feet away from the router and is up a level. Is it best to put the router in that room or put it about half way?

half way if you can and then make sure your router is using a WiFi channel that is not used by neighbors. This information can be found under the WiFi configuration in the router

Questions & Answers: ASUS RP-AC1900 AC1900 Dual Band Wi-Fi Range Extender

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ASUS RP-AC1900 AC1900 Dual Band Wi-Fi Range Extender / AiMesh Extender for Seamless Mesh Wi-Fi; Works with Any Wi-Fi Router

Dual band 802.11AC - utilize 2.4 GHz and 5 GHz bands to provide the best connections and speeds up to a combined 1900 Mbps
Asus AiMesh technology - connect with other AiMesh compatible routers to create an expansive and seamless Wi-Fi experience.
Maximum flexibility- suit your needs by using the AC1900 as either an AiMesh Node, access point, media bridge, or as a Wi-Fi extender that works perfectly with all routers.
Mobile app support - Control every feature at your fingertips using the Asus extender app or the Asus router app for AiMesh networks.
Network wide Features - access to a variety of features including AiProtection, parental controls, and adaptive QoS when connected to an AiMesh network.

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