There is no problem connecting wires to both the "back stab", and screw terminals of a receptacle. As long as the terminals are rated for the size of wire being attached. For example. Most "back stab" terminals are rated for 14 AWG solid copper wire, whereas screw terminals are usually 12 or 14 AWG solid or stranded copper.
With that said... It sounds like your receptacles might be hooked up backwards. The black ungrounded "hot" conductor should be connected to the brass colored screw, which is on the side of the receptacle with the shorter slot (at least in the US).
The red lead is wired to hot on BOTH the load and lead terminals of the GFCI. Likewise, the white lead is wired to neutral of both the load and lead terminals.
This sounds insanely wrong, and if it is in fact an Edison circuit -- two hots, shared neutral -- then a GFCI cannot possibly work right because the current on the neutral will not be the same as the current on the hot being measured by the GFCI. It will trip frequently.
I don't like shared neutral circuits at the best of times because remember the current on the shared neutral can be as high as the sum of the currents on the hots. Just because the neutral has no voltage does not mean it has no current, but the overcurrent protection on the breakers is on the hots. You don't want to get into an overcurrent situation on the neutral because nothing stops those wires from overheating.
Is this wired correctly, and if not, how do I rewire it so that it is?
There was a similar circuit in my house when I bought it and I ended up just pulling new correct wiring through the walls and doing the whole thing over. I concur that when you are rewiring an old house, everyone who lived there before you was dangerously ignorant; I've found terrifyingly bad wiring in GFCIs.
So, solution one: rewire everything so that there is one circuit for the GFCI and everything downstream that you want protected by it, and another circuit for the non-protected stuff.
However, if you don't want to actually pull new wire through the walls and do it all again, the next best thing to do is solution two: get a two-pole GFCI breaker and replace the breaker in the panel. Now you can throw away the GFCI outlet and replace it with a regular outlet. The downside is (1) expensive, and (2) when it trips you have to go to the panel to reset it.
I just replaced the outlet by splicing the outlet inline i.e. both the incoming and outgoing are connected to the hot lead terminal, same with neutral.
That's solution three, and it will work, but whoever wired it up originally might have wanted things downstream of the GFCI to be protected by it. This solution breaks that property.
Best Answer
I think I understand what you are asking.
My understanding is that you are not needing this light to be protected from the GFCI receptacle (since it wasn't before). You would just like to add a GFCI to the existing circuit.
It sounds great that you have figured out that the wires going to the light are "line wires". However I think you might be using that terminology a bit different than most electricians would. To us "line wires" are generally wires that are constantly hot. As in not switched power. This being a light I presume that the hot wire is actually switched power. Do you need to turn on a light switch for these lights to come on?
So the question is. Is it switched power going to your light? If the black wire going to your light is switched power, is the red wire switched power? You might need to trace back to where that red wire is fed from to see if you can get constant power from the previous box on the red wire to bring to this current box.
Basically you will end up with one of two scenarios.
Scenario 1 - Either the GFCI will only work if the light is on (because the hot is switched). Which you can still wire up. You will take the black and white "feed" wires in your box, and attach two pigtails on each wire. One of the pigtails will go to the light, and the other one will go to your Line side of your GFCI (for the black and white wires). You do not need to put all the cables on the GFCI or use the load side. Since you don't actually need anything protected by the GFCI.
Scenario 2 - You are able to get in your box a "switched hot" a "constant hot" and a neutral. You will wire your light up like before, and take the new "constant hot" you might have just found and put it on the line side of your GFCI and the neutral and also put on the line side of your GFCI.
Enjoy