What you propose should solve your dropout problems. The important setting in the Express is to have it connect "via Ethernet" and to configure its "Connection Sharing" to "Off (bridge mode)".
See if you can find these settings in AirPort Utility 6.1 - I prefer the more useful AirPort Utility 5.6, so I don't have a screenshot for you.
Sure, here you go.
Make the Express look like this:
If you also want the Express to serve as an additional access point (optional) set its Wireless Mode to "Create a wireless network" and provide the exact same Wireless Network Name and security settings as your existing router's wireless network.
Something like the following: