Yes, you can do this with EE, and I’m positive there are a few different ways to handle it. I tend to go with using separate weblogs for each of my data pieces - so I’d set up an “Events” weblog to capture and display the relevant events data. So, that’s probably a good first step: decide what weblogs you’ll need to create to both capture and spit out your data.
When you say “sign up to events”, do you mean: “Log in and submit events”? If so, you’ll probably want to examine using a SAEF for this. That way, users will be able to go to something like a “submit an event” page and they’ll be able to enter their own events and you can eventually control how they’re approved and displayed.
So, then the rest of this is easier. You posted “Have a full view of events”. I achieved something similar by using the Repeet plugin to display my events. I’d imagine you’d probably be able to use the same method to use on your front page in a sidebar or something - you’d just have to limit the amount of results you want to display.
I’m sort of generalizing here, but you really can create an events calendar with EE. If you search the forums for “calendar” and/or “events”, you’ll see some really great threads that should help light the way for you. Trust me, I’ve been through quite a few of them and started a few of my own. Earlier in the year, I was working on a personal project that never launched, but I was trying to come up with a nice way to display upcoming concerts at a listing of venues - both the concerts and venues details were to be user contributed. Users would be able to view a list of all upcoming concerts as well as be able to view each venue on a separate page along with that venue’s upcoming concerts.
I got it to work, but I can’t give away all the secrets 😉