If it's possible, should I accept such emails from users and what problems to expect when I will be sending mails to such addresses?
Not yet. The IEEE plans to do this: H-Online article: IEFT planning internationalised email addresses
I would assume yes since a number of top level domains already allow non ascii characters for domains and since the domain is part of an email address, it's perfectly possible. An example for such a domain would be www.öko.de
short answer: yes
not only in the username but also in the domain name are allowed.
The problem is that some mail clients (server-tools and / or desktop tools) don't support it and throw an 'invalid email' exception when you try to send a mail to an address which contains umlauts for example.
If you want full support, you could do the trick with converting the email-address parts to "punycode". This allows users to type in their addresses the usual way but you save it the supported-level way.
Example: müller.com » xn--mller-kva.com
Both points to the same thing.
The answer is yes, but they need to be encoded specially.
Look at this. Read the part that refers to email-headers and RFC 2047.