Can the default language change to el-GR or not?

CBs default loaded language is entirely based off what Joomla's default language is. You'll need an el-GR language plugin for both CB and Joomla then mark it as default in Joomla.

When I create a new field, should it be in English and then to translate in Greek or in Greek and then to translate in English?

Entirely up to you. It's best to not use language specific strings at all, but to use language keys. Example as follows

Field
Title: This is my field
Override
Key: This is my field
String: This is my field TRANSLATED!

vs

Field
Title: MY_FIELD_1
Override
Key: MY_FIELD_1
String: This is my field TRANSLATED!

If using language key approach you'll need to be sure each of your installed CB language plugins has an override for that key.

Either one or the other (question 2), where should I add the translation? (it doesn't appear in overrides).

Add it manually to CBs language overrides. You don't have to use language strings found with the language finder. See the below if unsure how to use CBs language overrides feature.

www.joomlapolis.com/blog/kyle/18701-language-overrides-have-arrived

The default language in Joomla is el-GR and if I understood correctly what you have typed it should show CB->Plugins->Default language (Greek) but it shows CB->Plugins->Default language (English).

The default language in CB is just a plugin name. It's not going to rename your other language plugin and say it's default. CB doesn't determine what language is loaded. Joomla does.