Learn Croatian: To je pravilo, bez obzira na iznimke! - That is the rule, regardless of the exceptions!

It literally translates to 'without regard to' or 'without consideration onto'. The noun obzir means 'regard' or 'consideration'. Because it comes immediately after the preposition bez (without), which strictly requires the genitive case, obzir takes the genitive ending -a to become obzira.

What case is the word iznimke in this sentence?

It is in the accusative plural. The phrase bez obzira na always ends with the preposition na, which forces the following noun into the accusative case. The singular noun is iznimka (exception). For feminine nouns ending in -a, both the nominative and accusative plural forms end in -e, so it looks identical to the standard dictionary plural form.

Can I use bez obzira na before a verb clause, like 'regardless of what happens'?

Yes, but you have to bridge the preposition and the verb using to što (the fact that / that which). You cannot put a verb clause directly after na. Instead of just attaching a noun like iznimke, you would say something like bez obzira na to što se događa (regardless of what is happening).