If you were to draw a person on a Minkowski diagram, that was unmoving (though time was still passing) what would that look like? (As a light cone and ignore the observer.)
Also what would it look like if you drew a person looking into a black hole (at a safe distance) but NOT accelerating away? (also as a light cone)
I'm not sure what you mean by "as a light cone". An point-object's path through space-time is represented not by a cone, but a curve.
I can probably help you on your first question. First, pick a position for your person at time zero. Plot this point on your space-time diagram. Then, choose a (short) time later. Plot the position for your person. (Did the position change?) Keep doing this. Connect the dots.
If the object is standing still his world line is vertical in minkowski diagram. It doesnen't matter if he is observing a black hole or an apple...
After taking a closer look at your image I saw that the cone you drew is colored yellow. For the sake of simplicity lets consider a 1D example (we have spatial dimension "$x$" and time timension $ct$) where the "light cone" becomes a "light X". "Light X" consists of two asymptotes angled $45^\circ$ and angled $135^\circ$ according to the spatial direction $x$. It is important to realize that:
only the objects which travel with the speed of light can exist on the asymptotes of "light X" and that theese objects can only moove in the direction of asymptotes mentioned above or paralell to them.
Now tell me if you need an extra explaination on how the "light X" is derived and why asymptotes are angled at $35^\circ$ and $135^\circ$.