Well, at least in my experience with EU projects, they tend to be much more restrictive with data sharing than equivalent US institutions: e.g. a lot of paid EUMET data has publicly available NOAA equivalents - though usually of worse quality.
Depends on which model. Only really the ECMWF weather model is not fully free. The German, French, Dutch, ... models are all free (regional and global models). Of course, these global models are generally less accurate than ECMWF, still ECMWF has a lot of free data available too. US models are also freely available, and quite easy to work with (as opposed to some European ones).
It is not an EU project. It is an ESA and EUMETSAT project. Neither is an EU organisation. Both have multiple non-EU members, and I do not think all EU countries are members of either.
https://user.eumetsat.int/resources/user-guides/getting-star...