I’ll probably regret asking this so please excuse the ignorance, never had to do this, but …
Do I need a headset press in order to fit Chris King or Hope Head Doctor type headsets ?
Or do they just push in by hand ?
And how do I get the crown race thing off of the fork steerer tube ?
And what about one of those flared hollow tools to extract the old one, do I need one of those ?

The flared hollow tube headset removers are a good idea to use yes. In my experience they cause less damage. To install a headset you can use a flat block of wood and a hammer and some patience making sure you knock it in straight and level or Google homemade headset press. It’s very cheap and easy to make and use.

If they push in by hand then something is wrong.
A head set press is a tool with some nicely shaped press parts that allows you to wind the headset into place and to apply pressure evenly. You can still get it wrong with a press but it’s harder.

The home made versions are easy with threaded bar etc but getting the right washers might be harder.

Smacking it with wood and a hammer works but, you can still get it right but it’s harder 🙂

Personally I wouldn’t start knocking a chris king or hope headset with a hammer & wood as the headset will outlast the frame is you do it right. I’m assuming your ordering the headset from the internet? YOu can always check your LBS for installation prices, many will fit if you buy it from them.