If you are working in another country, you and your family must be treated exactly the same as your colleagues who are nationals of that country. This applies especially to:
Equal treatment prohibits not only open discrimination, but also any rules that indirectly place you at a disadvantage by, for instance, limiting your freedom of movement.
For example, a rule requiring you to live in the country for a long time before you can access a particular public service there would be illegal.
YES — Whatever their nationality (even non-EU), your family members:
If you think that a rule or decision affecting you is against EU rules allowing you to work abroad, you must first take your case to the authorities in the country where you work.
Depending on the exact problem, you might also be able to get help from the EU assistance services.
YES — If he's recognised as having the right to stay with you in Slovakia, he'll enjoy the same rights as you.