Is there a way to set up the AE at home, before I travel, and saving the setting as a profile so that I can simply "plug & surf" when I get to the other side?
Yes. The AirPort Express Base Station (AX) can store up to five different configuration profiles. You will need to use the AirPort Utility to create & save these profiles.
To create a new profile:
o Open AirPort Utility, select your AX from the list, and then, click "Manual Setup."
o From the AirPort Utility menu, select "Base Station," and then "Manage Profiles."
o Click "+" to create a new profile; give it a name, and then, click OK.
o Set options, such as network name, passwords, Internet connection method, etc.
o When done setting the options, click Update.
To switch profiles:
o Open the AirPort Utility, select your AX from the list, and then, click "Manual Setup."
o From the AirPort Utility menu, select "Base Station," and then "Manage Profiles."
o Click on the desired profile, and then, click OK to apply the new profile and restart the AX.
Just remember to switch to the desired configuration profile before removing power from the AX.
Bear in mind that every hotel has an authentication process when you connect either wirelessly or by ethernet cable i.e I plug in the ethernet cable, launch Safari and get the hotel's login page.
Once your AX is providing a basic wireless network, your iPad should be able to reach the hotel's router at which point you would typically get these "authentication" pages within the web browser. These are not something that would necessarily be unique technically ... only the login credentials would be different.
If I was to simply plug in the room's ethernet cable into the AE and launch Safari on the iPad/iPhone, would the same thing happen?
Pretty much. As mentioned in my earlier reply, if you reconfigure the AX as a bridge, it should accommodate most situations that you will encounter. In this configuration, the AX will "pass-thru" the information from the hotel's router and be broadcasting its own wireless network for your iPad to connect to. One thing you may want to do, is to enable wireless encryption so that uninvited "guests" won't be able to take advantage of your wireless network.