How to keep a GFCI outlet with tight clearance from shorting inside a steel electrical box

Ok I read your question a few times, so hope I understand what you want to do. You want GFI prtection on the outlet , but no GFI protection to the fridge. This will be easy to do if you can affirm that the load wire leaving the j-box where the switch and outlet are located, is actually the feed for the fridge. To check this out, you need to turn off the power, check the outlet at the sink and fridge to be sure they are ,in fact on same circuit and off. Now disconnect all the wires from the outlet and any wirenuts so everything is isolated. Now, carefully turn the power back on and check the hot (typically black) leads to ground with a volt meter to determine which one is the feed/source wire. Mark this with some red electrical tape. Double check to see that the fridge outlet is still dead.

Next, turn off the power and wire nut the black source wire and associated white neutral to the black and white wires you suspect goes to the fridge.( black to black, white to white) Turn the power back on and check with your voltmeter at fridge outlet again. If there is voltage there now, you have found the right feed wire to the fridge outlet. An alternate method of finding that wire with the power off, is to use an ohm meter. Assure the power is off, then twist the black and white together on the wire you suspect goes to fridge and check the hot and neutral slots of the fridge outlet with your ohm meter. the meter should show 0 ohms or "short circuit".

Now that you have identified the hot feed and load wire to fridge in your box, you can wire it so only the counter outlets are GFI protected. Put the source black wire together with the fridge black wire, along with a separate 8 inch piece of black wire (pig tail) and wire nut them all together. Use the 8 inch black wire to feed your switch/gfi hot. The neutrals tie together as usual with an extra pig tail for your GFI outlet neutral. Obviously, trim the pig tails to a comfortable length to fit in your box before connection to the GFI.

Since all outlets must be GFI protected in the counter outlet and since you cannot split a gfi outlet top and bottom like in your diagram, you have to do your light differently from your previous plan. I would suggest using a switch/single outlet device wired from the load side of the gfi. Wire the switch in series with this single outlet. This means only the single outlet is switched and gfi protected. You must have gfi protection on this outlet, as someone could unplug the lights and use it for something else.

Hopefully, one of my artistic buddies can do an edit and add a nice diagram depicting what I have outlined for you.

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.