What is simpler, to use ethernet connection or wifi?
Ethernet, but not by much. It also depends on if you have am Ethernet connection nearby already. I would say that if you had to run wire through your walls, WiFi would be easier.
Should I / Can I use it as a server or is better to use local web server for the site and the device just as external interface?
You can. Whether or not you should will depend on how robust you want this web interface to be.
What shall I use as storage, Arduino, clearly has not enough memory;
What makes you say that?
shall I use an SD card or other kind of external memory? How much memory would I need for <10 remote controls?
Do these remotes have 50 buttons or 3? If the former, do you plan on storing all the commands for every button? I know there are buttons on some of my remotes that I never bother pushing.
Is there any arduino library that I shall familiarize myself with?
IR remote library seems relevant.