There are lots of things that can happen when using a VPN like a those you suggest in your question. It would be very difficult to list and define a resolution for each item. Another item that could create problems is a VPN may use ports that the remote network does not permit through their firewall. The VPN could be using IPSEC or SSL. There are different steps to troubleshooting each type of connection. It might be easier to post issues as you encounter them or do a Google search for “vpn troubleshooting” or “vpn tutorial”. I think you might find your answers in this type of research.
The local firewall should have already been configured to support VPN. If not, take the time to do that before traveling to the remote location.
The proxy can be turned off (if permitted) by the user by modifying their browser settings. They will need education on how to do this.
Last time I was in Vegas I wasn’t able to VPN in from the hotel. The hotel’s network wasn’t configured to allow for Cisco VPN (which we use) which was preventing me from VPNing in. There network was blocking UDP port 500 which is used for the first stage.