a) No, the wi-fi is internal to the phone. Unless the phone has an Ethernet slot built into it (unlikely) you simply use the existing wi-fi network.
b) Are there actually many places that enable you to plug in your own Router? I would think that most places would already have wi-fi (like hotels) and would give you the code to punch into your phone or laptop to access the secured wi-fi network.
If a hotel room offered ETHERNET only, then there would already be an Ethernet slot (probably next to the phone jack) which you would need to plug into your device.
Actually, the ROUTER plugs into a phone jack. If Ethernet is offered then you should already have the Ethernet jack available. If so, you want an Ethernet SPLITTER to support multiple Ethernet devices using the same Ethernet connection, not a ROUTER.