speaking as a single mom here who has dealt with many of the same issues:

Yes, you can bring the child in with you, provided you are listed on documents as his legal or birth father. If you do not share family name you will need to make sure you have ample documentation of your relationship, as not sharing a family name is rather uncommon here.

Sponsoring a nanny simply requires paperwork that indicates that you are either married or divorced. That said, I am unsure how the authorities would perceive it for you (a man) to sponsor a live-in female (have you considered hiring a "manny" instead?). Don't know of any man who has done that, and my experience with single men who have wanted to sponsor/hire even a housekeeper here is that it has been shot down, specifically because it would mean a single man living with a single woman...

It SHOULD be do-able, but folks here often can't get past the male-female stuff....

I would think you might be forced to either choose a part-timer (non-live in, non-sponsored female nanny) or else hire a male nanny.

"Most plain girls are virtuous because of the scarcity of opportunity to be otherwise."
-- Maya Angelou