I think the first thing I would attempt to do would be to adjust the rod such that the pot assumes a circular shape once again. Regarding the skin tension and action.

A plastic Remo head will take quite a lot of tension and tightening it will tend to raise the action a little. It should be tightened by tightening each of the tensioners a little in a clockwise or anticlockwise manner (i.e. not "opposites" although this might seem logical at first"). The feet should not "sink" appreciably into the head. You should continue to tighthen the head until you get a nice percussive sound (a bit like a snare drum) from tapping with your fingernail. This may need a couple of trips round the tensioners adjusting a little a time. You may find that this raises the action enough to prevent the string fouling you mention.

If the neck is reasonably straight, there are then two remaining ways to raise the action, one by adjusting the pot angle in relation to the neck (if the banjo has such an adjuster) or by using a higher bridge. Frequently a tiny raising is all that is necessary and this could be achieved by gluing a thin strip of hardwood veneer under the feet of the bridge and trimming so this is the same size as the feet and therefore virtually invisible.

These are typical banjo adjustments and, not knowing the model in question, I couldn't answer some of your other questions, e.g. the role of the Philips screw.

If more elaborate work is necessary, I would suggest taking it to Andy who I have used in the past and who is excellent.

Peace

moo