Menus, categories (and sometimes items) need to be properly configured to the desired language.

If you use common K2 categories for content in both English and Greek, you're doing it the wrong way. Separate your content into different category trees and properly assigned them to the right language.

You can't use Falang and Joomla's multilingual features at the same time. You can just use the latter.

Use this guide to set things up: docs.joomla.org/J3.x:Setup_a_Multilingual_Site

Not really, sorry. Falang stores the translations in its own database tables, it doesn't map an item assigned e.g. to English with another that is its translation for German for example.

I actually have a plugin for that in the works. Use the contact form here www.joomlaworks.net/support/get-help/contact if you'd like to get a (paid) preview before it's released for everyone.