To set up the internet on your PS3 first make sure that you have a wireless network with a internet connection or have an Ethernet cable from a router with internet plugged into your PS3. Next go into Settings>Network Settings>Set Up Connection. Follow the instructions on screen making sure you input valid information. Submit and your PS3 should now be connected to the internet!
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If you laptop has an HDMI output then you can hook it up directly to a HDTV with an HDMI input without the PS3.
If the hotel has broadband WiFi you don't need the computer just use the PS3 WiFi connection and if it has Ethernet then plug the Ethernet cable in your PS3.
The back of the PS3 has a spot for an Ethernet cable to allow wired connection with the internet. Purchase a Cat 5 or Cat 6 ethernet cable long enough to go for the back of the PS3 to the source of your internet connection. Ebay had Cat 6 50 foot cables for under 20$ including S/H.
See related link for help setting up the PS3 internet connection
Yes depending on the model that is used.