Engine Egr removal

The emissions laws require that all original emissions stuff on the car be left intact, but since you are putting in an 'EGR type' of intake manifold into a car that never had that; it wont cause any problems with the check engine light nor the codes. That is assuming the EGR valve is not leaking air into the intake manifold.
Thats the good news.
The bad news is that the sticker on the hood may show the non-egr manifold and its non-egr vacuum hoses, and a safety or emissions inspector might object to the apparent discrepancy of an intake manifold with either an EGR valve not connected to anything, or a blanking plate to replace the EGR. That would be a really vigilant inspector.

To be totally safe, keeping the non-egr intake manifold would solve that problem. The fuel rail could be swapped, since its easier to let it remain in the non-egr manifold.
If you can get the egr fitting unscrewed from the 96 exhaust manifold, just plugging the hole with a pipe thread plug of the right size should do it. ......But a super vigilant inspector might wonder what was originally connected to that plug in the manifold. As I mentioned the emissions laws require that all original emissions stuff be left intact.
Hopefully we never get emissions inspectors that would know to look for the two tiny bolt holes in the back-side of the bracket under the coil pack - that originally braced the side of the EGR air tube!!

Im curious why you are swapping a 92 engine for a 96 engine, since they are the same internally and horsepower-wise. Is there something wrong with the 92 engine? My route would likely be to pull the 92 out, do a rebuild and put it back. (It would go back in with [at least] the 5 new freeze plugs and an engine head rebuilt with newly installed intake valve seat inserts; and anything else it needed for renewal). The one larger freeze plug on end of the Escort engine blocks can only be replaced when the transmission and flywheel are out of the way.

It can be hard to get the CV axles to go back into the trans all the way. They have a spring inside the inner CV casing whose purpose is to take out any 'play' as the driveline and suspension go around corners and over bumps. You need to push in firmly on the end of the CV axle (where the hub nut goes), and smack it with a hammer. It wont hurt the threads at all, as they are back-spaced. Then you might be able to see the inner casings all the way seated into the trans. The gap is perhaps 1/4" or less.

My Escorts are all the LX kind, and I dont know how difference the DOHC enginges are. Anyway; I always prefer to hoist out the driveline, with eng. and trans. still bolted together. If the engine is still in place, auto trans and the manual trans can only come out downward. I dont like to do this because of the difficulty of getting the trans lined up again and shoved into place -upward. The engine can come out upward if its an automatic. With a 5-speed the long input shaft prevents the engine from coming far enough out of the trans to be able to be hoisted. Hence my preference, to hoist both engine and trans together.
You have to disconnect the cv axle outer joints (the Birfield-Rzeppa joints), and pull the CV axle assembly out out of the trans, before anything can be hoisted out. This is also when I check that the ball joints are not loose - or even 'nearly loose'.
Good Luck.

If you have one cv axle that is not fully installed, you can pull it out to look in there with a mirror, to see if the splines inside the gear are lined up with the hole where the CV axle goes in .
I like to put a chunk of PVC pipe into the opening after I get the CV axles out of the differential. That will keep them from dropping out of place. That said; I have worked on 8 Escorts axles, and have yet to have any gears drop out of place.
If that happens, it requires the trans be removed, and opened to to where you can split the case halves. Then you can put the diff output gears back into place.

You only have to hold one axle gear in place - because it will prevent the opposite one from moving out of position. There is a tool useful for getting cv axles out of the drivers side of the transaxle; an OTC-7140. I got it from ebay, for about $40. But for $39 less than that you can use a 6" chunk of broom handle of pvc pipe.

I have run into the matter of the "C" clips being of a thicker metal. I wish there was a source for the correct size. As it is I usually replace the CV assemblies once I take them out - due to worn or torn boots.

When you say it wont go into any gear, can you move the lever into the various positions - with the engine running? - and not have the wheels spin? And is the car still in the air, so the wheels can just spin in gear? Even in neutral they might turn slowly with the engine running.
It should at least 'grind' a little when the shift lever is pushed to the right and pulled back for reverse. Im wondering if the slave cylinder is merely not pushing the release fork down far enough for the throwout bearing to be pushing on the pressure plate - so it comes to a halt when you have the pedal pressred to the floor. (Total bleeding of the mastercylnder and slave cylinders takes patience!)
I only now about the LX escorts, so measure my advice based on that.
There is a shifter shaft sticking out of the bottom rear corner of the 5-speed trans. It gets connected to the long 'shift control rod' at the double bushing, (Ford name for it), and moves when you put the shift lever into gear. It connects to the shift control rod via a small bolt that goes through the shifting shaft and the front end of the shift control rod. A small rubber bellows covers this up, and to get the bolt undone for trans removal, or reconnected after trans reinstall, you have to force the bellows to one extreme so you can see the bolt.
If you are under the car (with engine stopped), you should be able to slide the shift shaft fore and aft, and slightly rotate it (using pliers). Being twisted toward one side and shoved forward (toward the front of the vehicle) would be putting the trans into second or reverse. Moving the shift shaft forward without forcing it to one side (its spring loaded) would put the trans into 4th gear.

I hopy you were able to get the VSS out, so you can check the fluid level - and recheck is within the next year.

The shifting shaft that sticks out of the transmission connects to the shifting control rod - and the joint is covered by the small bellows. At least that is how my two 5-speeds are set up.

The rod visible by looking into the differential section of the trans is what keeps the differential bevel gears in position; they arent visible, but they mate with the side gears having the internal splines on them - which the CV axles mate with. Its normal, and is not the reason you cant get the trans into various gears - unless its rusty.

When the CV axles are removed, the gears they slid out of (the ones with the internal splines) can move out of place. Not that its a likelihood, but if they move far enough to drop out of place, the only fix is to remove the transmission, unbolt the two halves of the case, and put the gears back into correct position. It cannot be done by working through the hole in the gear, because the gear would have dropped out of sight! (and believe me, I have tried!) That is why I always stuff a short chunk of broom handle or a piece of about 1" pvc pipe into one side of the differential, and hold it in with duct tape. It wont matter which side you put the 'plug' into, as keeping one of the gears in position also keeps the other one from getting out of place.

It is possible to push out the 2nd CV axle inner joint from the open end where the first cv axle joint was. But it means you have to work past that rod. This may be possible with a very long screwdriver of -just- the right size, but the OTC-7140 tool makes it much easier. This tool is made with one end having legs on each side of a central slot, that are long enough to go beyond the rod, and bear against the innnermost end of the 'other' CV' axle assembly. Then one good smack with a hammer, and the other CV axle will have popped out about 1/4" - and from there you would be able to pull it out by hand. Maybe you were lucky enough to get the driver's side axle out somehow. ?

As for putting the clutch in wrong, there are only two ways the clutch disc can face; the correct way, or backwards. The clutch disc has a raised center part holding sideways springs, and that raised part must face toward the pressure plate. Otherwise the raised section would be squashed against the flywheel, and not be able to be dis-engaged. It may be possible to have installed the release fork incorrectly, such as with the balance point not seated into the ball sticking out of the bell housing.

I would still be thinking the problem could be a transmission that was out of fluid. If you have the trans out, its fairly easy to unbolt the sheet metal item at the driver's side, and see if there signs of fluid in it.

And by the way, the amount of stuff involved to remove/replace the dashboard and heater core on an Escort makes it a big job too. I have done that to two of mine, both times to replace heater cores. One had begun to leak, the other one was done so my daughter would not have to face that repair job herself. She lives in western VA, and we spend winters in far south TX, 8 miles from Mexico.