I don't have a good level of mathematics.
So you won't have much GR either, then, and you are trying to find out about the space-time geometry inside a black hole?
Have you gone through the basics with non-rotating, neutral charge, black holes? (References below JIC.)
I don't think I can give you a complete explanation in one go. Instead I'll try pointing you in a useful direction.
Loosely. What the article is referring to is a method of doing a calculation called a "perturbation" where you start with the well-known solutions and then tweak the situation a little - i.e. you start modelling the Earth as a sphere, then gradually account for how the Earth diverges from perfectly spherical after working out the results for the simpler shape. The calculation usually involves a lot of steps - think of an infinite sum. This is not a problem when the terms get smaller and smaller in a nice way so the overall sum converges, and this is what normally happens like when you want to figure the gravity of the Earth. However, if you try to use this method for specific types of black hole, the perturbation calculation no longer converges everywhere. You can think of the incoming and outgoing "streams" as an artifact of the calculation method. The "instability" just means that the calculation method fails.
Bottom line is that this is one of those things where you need the maths.
Per your specific questions:
What happens to radial and time directionsr and t inside different black holes?
Time and space swap roles across the event horizons
http://www.einstein-online.info/spotlights/changing_places
http://www.jimhaldenwang.com/black_hole.htm
... because of the symmetry, it is useful to define a radial direction and a couple of angular ones for space.
What is a black hole 'firewall?'
Firewall: have you seen:
http://en.wikipedia.org/wiki/Firewall_(physics)
And can the singularity repel stuff out rather than suck it in?
I suspect you are being glib here, but the key to figuring out relativity is to be careful with how you say things.
The "singularity" is just the name given to the situation where our maths fail - it has no physical meaning. It's just a mathy way of saying "we don't know what's going on here". You can see this with just normal gravity ... the force is given by ##F=GMm/r^2##, and you can see that the force blows up at ##r=0##. It just means that the equation does not work when the small mass m is very close to the center of the big one M. Same with the singularity of a black hole.
"Repel" and "attract" are terms that make no sense in space-time ... these are concepts that require a well-defined time axis. Inside the black hole you don't have one, so the articles are playing a bit loose with the terminology. If you've followed the basic stuff above, you may have a better shot at figuring what the authors you are trying to understand are trying to talk about.