do you think it is possible to know everything about a programming language?

To a certain extent. I think you can regularly use all of a relatively small language like C. But the base language isn't the most important part; the functionality of standard and 3rd party libraries is much more interesting. The basic syntax of Perl is interesting but CPAN is where Perl really shines. Knowing which modules to use for certain kinds of problems can be much more helpful than knowing everything about Perl's syntax.

Computer languages are tools for solving problems. Some people see the language itself as a subject for exploration, and others just need to get a job done, preferably as fast as possible. The latter type of people are just interested in the end result and don't really care if it can be done more elegantly.

are day-by-day problem-solving tasks a limitation on the things that you can learn about the language?

Not necessarily. If all you do each day is write the same type of programs, then learning some new features isn't really necessary. However, I think that being a programmer means that you're allergic to repeated tasks, and look for ways to automate those away. It's that search for automation which almost forces you to learn new things. Of course, lots of people are happy with using a fixed subset of a language and have no desire to learn more.

how important is the presence of a community, real or, like perlmonks, virtual?

It always helps to have access to "real" people for asking questions, handholding, etc. Some people need that very badly, others are more inclined to search for answers to their questions by themselves.