4. no, the graph of gravitational field strength for a uniform solid sphere looks like this...
zero at the centre,radius zero (r= 0)
increases linearly between the centre and the surface (r<a)
maximum at the surface (r=a)
decreases between the surface and infinity following the 1/(r^2) law
stars and planets aren't exactly uniform spheres but it's close enough
a star doing it's steady business is in equilibrium - the pressure at the core is high but not high enough to make a neutron star or B/H.
the gravity acting inwards is in equilibrium with the radiation pressure pushing outwards, radiation pressure is just pressure on the stellar gas caused by radiation from the fusion in the core.
when the fusion stops due to the fuel being exhausted, the material the outer layers of the star are made of falls towards the centre under gravity and violently hammers the core making the pressure there greater than it ever was while fusion was taking place, only for a short time but it's enough. might be worth thinking about the difference between placing a hammer on top of a nail and then trying to push the nail into some wood and using the hammer properly by striking the head of the nail at some speed.
having the momentum of the hammer head build up over a great distance and then stop abruptly creates a much greater pressure on the nail than just pushing on it steadily.
note core collapse only happens in the largest stars - not stars like the sun.
the HR diagram is central to understanding stellar evolution - it's in the astronomy gcse. if you're not being taught it I guess it's not going to be part of your physics exam, but you could always teach yourself if you're interested e.g.
http://www.atnf.csiro.au/outreach/education/senior/astrophysics/stellarevolution_hrintro.html