Does it require you to consistently keep Neutral karma? Or only when the option to take the perk comes? If your karma does lose neutrality then does the perk get inactivated?

Also how do I check my karma? Can you check the exact numbers?

No, the only impact karma has is on qualifying for the perk itself, and in the case of the good karma perk, how much extra HP you get when you get it.

You can check your Karma by going to the general tab of your pip-boy and clicking whatever button says Karma.