It's a low fuel sensor, if you don't have a warning lamp for low fuel, don't worry about it, it will still work as it is supposed to.

Anything is possible, but is it a good idea, no.
Look at GM's Olds diesel conversions. They were also a good idea but didn't work too well in the real world.
To do it right, it needs to start out being designed and built as diesel.
And to add a personal opinion, the A series is a great engine family for a gasoline engine, but I think that if one were able to successfully convert one to run on diesel, the stresses would cause one to self destruct.

Try this, put a fuel filter between the fuel pump and the carb. If this fixes the problem your fuel pressure is too high. If it doesn't fix the problem look at the needle and seat gasket it could be leaking fuel, or the needle and seat is loose.

Actually, the differences between 4 and 5 speeds internally is behind the center plate where 5th gear was added. The front half (bell housing) differences have to do with which engine it bolts to, as in bolt pattern, etc.
A lot of the main shaft gears interchange between a given series of trans, it's the counter shaft (lay shaft) that's different because of 5th gear, but again this all lives behind the center plate.
Given this, IF you have an A series 60 series 4 speed, and an L series 60 series 5 speed, you should be able to put the bell housing from the A series in place of the L series bell housing and make an A series 5 speed out of it.
Don't quote me on this because there may be differences that I don't know about that could make this not work, BUT from what I remember from when I worked at a dealer it SHOULD work with few issues.

My point exactly, WHY DID IT FAIL?, Until we know that, we can't help you.
Mechanics are like Dr.'s we all think that we're right and everyone else is wrong, this is why I suggest that you find another one. Just like a Dr. get a second opinion.
This guy may see you as someone they can buffalo and blow smoke up your backside.
Rallytwit is right they did come with an air pump and no converter, even in California.
Look at the emissions sticker pasted on the bottom side of the hood, that will tell you what emission equipment it needs to have.
Find out why it failed first before you spend any money.

The big question is "why is it failing the emissions test?". Is it high CO, high HC, or something else? The first thing you should do is get a can of SEAFOAM and add it to a full tank of gas, then go for a very long drive. After that, change the oil, change the spark plugs, and get the valves adjusted correctly. (many people don't know that valves need to be adjusted at regular intervals.) Then go and get it tested again and get a copy of why it fails, if it does so that you can see what it failed for. And above all, maybe you should consider finding a different mechanic.
Personal opinion, the valves are probably in need of adjustment, and your mechanic thinks that it's something worse.

Something else to check is the ignition switch, if the internal contacts are worn too much, it could cause you to get low or no voltage. Use a volt meter and check voltage in both the crank position and the run position, both should read 12 volts at the switch, and at the coil during crank, and at the resistor in the run position.

Bad Idea.
Find a way to properly vent it so that it goes into the filtered side of the air filter so that the fuel tank will can vent. This is where the fuel tank gets an air supply so that it won't collapse as fuel is pulled out of it.
Just blocking it off could cause you to have fuel delivery issues.

Starting with the second photo, the hose with the red square is a fuel vapor vent hose and should go to the filtered side of the air cleaner.
The green circle is an engine vacuum port that could be used for your power assisted brakes if you can locate a fitting that the hose will fit onto and still screw into the manifold.
The red circle in the first photo is a vacuum fitting, what you need for vacuum advance is ported vacuum. To check for this run the engine at an idle and test for vacuum there should be none, but when engine speed is 1500 to 2000 rpm you have vacuum this is ported vacuum.
The reason why you want it this way is so that you don't have full vacuum advance at idle, and so that the advance starts to increase as engine speed increases.
Hope that this makes sense.

Or you could have a competant machine shop destroke the crankshaft, but then you would run into all sorts of issues like conrod length and piston pin height.
Either way it could get real expensive.