Well...
I managed to go for a short 3-minute drive around the corner and back, at slightly above idle, without any problems/leaks/overheating. I attempted to record it, but my $40 "action sports cam" didn't save the file, even though it showed it was recording. You're honestly not missing anything exciting though, probably the most excitement was that the car shifted into 2nd coming back down my street. I've now confirmed 1st, 2nd and reverse all work! Hey, at least that should be good enough for autox. 
between you and mazduce, this forum is kicking project car ass!
Video or it didn't happen.
J/K....WOOHOOO! THE BROWN BOMBER LIVES!!!
Stampie
SuperDork
5/25/17 8:42 p.m.
If your drive was longer than 1/4 mile then you should be good to go.
Thanks everyone! I definitely couldn't have made it this far without the help and input of everyone here. Though I fully expect more frustrating problems and failures will occur as I continue to push things further.
However, I need to make a confession...
I've been on GRM since the fall of 1998, and although I posted fairly frequently, I never posted all that often. So it seemed like a project car might be the way to bump up my post count, and apparently it worked - I didn't even realize I'd hit MegaDork 103 posts ago! 
And here I thought you just did this because C4 is so under appreciated and cool. 
In reply to Dusterbd13:
Definitely. So glad I found this forum. So many forums are dead from people going to Facebook.
zordak
New Reader
5/26/17 9:35 a.m.
Congratulations, It is always a good feeling when a project moves under its own power. Shows you can do something constructive and bring an old car back to life.
Pete, Huge Congratulations!
I know you've got a lot more work to do, but thanks so much for bringing us along this journey of rehabing this old car. btw, were you wearing your gold chain during the drive?
In reply to Indy-Guy:
I've not bought one yet...looks like I need to put that at the top of my Father's Day list. 
I declare the name of this beast.........PETE'$ DRAGON
Well E36 M3...

I need an intervention ASAP - I'm already turning into the stereotypical corvette owner!!! 
I did take it for a bit longer drive tonight, about 2-miles to the gas station, then another mile to Advance to show them the car actually moves.
Unfortunately as I pulled into their parking lot smoke started billowing from under the hood. I discovered the trans cooler line is leaking again - I think I need to re-flare the end - and a mysterious oil leak somewhere from the back of the block. My hunch is it's the oil pressure sensor, as the threads felt Alamo stripped when I installed it, although I couldn't actually see any oil. Since it's not being used I'll remove it & install a plug instead.
A couple observations from the drive: these Hoosiers are very flat-spotted, so it's hard to tell what condition the suspension & u-joints are in(they seemed fine by hand though). The car also tramlines all over, but maybe that's to be expected? There's a slight pull to the right too, but honestly I've not checked tire pressures since we mounted them...they're still round & hold air.
I need to grab my camera out of the car & see if it recorded anything tonight, though I'm not counting on it.
In reply to Ovid_and_Flem:
I just hope there's no fire spitting out of it!
Again...
Stampie
SuperDork
5/26/17 8:32 p.m.
In reply to Pete Gossett:
If they're flat spotted maybe a burnout will take enough off to make them round again. Just saying ...
I'm glad you decided that this car was the one to properly build thread. You've got a great mix of knowledge and curiosity, success and setback, that works for the forum. It's a good contribution.
In reply to mazdeuce:
Thanks! In retrospect I don't think I'd have been able to get this far if I'd not picked a Chevy(or possibly Ford/Mopar). I definitely wouldn't have gotten this far without the help of everyone here, and elsewhere, but part of that is also due to the millions of SBCs built and hundreds of thousands of C4 Corvettes produced, which has led to such a vast knowledge base to tap from. I've also been lucky that most of the Corvette-specific bits have been serviceable, while the other GM parts have been dirt-cheap for the most part.
Comparing the Vette to my Europa(which I never got running), both cars were in similar condition, but there were very few people with the expertise on the chassis or engine to offer much help. Additionally, where the L98 was ~$500 to rebuild, with a couple mild upgrades along the way, the Twin Cam was going to be ~$2500 in parts alone if nothing was bent or broken inside, all for a stock 105hp...plus machine work. Or the alternative of custom-engineering an engine swap, and all the cu$tom machine work that was beyond my skill(and likely budget) to complete. Oh, and that Twin Cam needs a valve job every 40,000 miles.
In reply to Stampie:
I've tried that before, it never worked out. 
Buy a triple A card and go for a 20 mile drive, If the tires are still Flat Spotted then you know what's next. do you have a tap for the Oil pressure hole, clean it and of course check the threads on the Switch, I've never seen that hole Buggered since the switch is aluminum.
Pete, I remember when that brown/bronze two tone color combo came out in the mI'd 80s and it seemed so odd on a vette. But looking at it now it has stood the test of time. Ypur hard work is gonna pay off....it essentially is a 60k mile survivor and looks period correct cool.
And let's not forget it was a radical car for its day.
In reply to GTXVette:
My plan is to get rid of the weird aluminum T that both the oil pressure switch and oil pressure sender thread into, and thread the oil pressure switch directly into the block(for now anyway). Hopefully that's the source of the leak.
I've never had AAA, but I might consider it once I get the confidence to take it out of town. I do have roadside assistance on my policy, but AAA offers way better services.
In reply to Ovid_and_Flem:
Ten years ago I would have never considered owning a brown or bronze car, but I do really like the color combo on the Vette. I think I'm turning into my father, as brown was his favorite color. Being a child of the 80's I was very much into the bright colors of that era for a long time, but not anymore.
I made a bit of progress in the garage this morning before it got too hot, and checked one of the three projects off my list. I removed the oil pressure sending unit, which was quite a challenge - the threads weren't stripped as I thought, but rather the top 1/2 of the unit was twisting on the base.

Fortunately they put a hex on the bottom of the base...unfortunately that's all suffed back here:

Adding to the difficulty, the whole assembly is trying to rotate in two axis...the two I don't want..while trying to unscrew the sender in the 3rd axis. Then there's the confined space to work in.
Here's how the whole setup is installed from the factory. The piece on the left threads into the rear oil galley on the top of the block, then the angled extension, then the switch and sensor.

I discovered that while the switch threaded directly into the oil galley, my Weiand intake obstructed it from seating fully, so I ended up using the the first piece above & threading the switch directly into in.
My next project is attempting to re-flare the trans cooler line, hopefully this evening. After that...
The more I think about it I'm going to keep the fuel pump wired to a toggle switch, additionally, I'm going to use one for the main cooling fan, and possibly add one for the coil. Eventually I'll add one for the trans lockup clutch too. So I'll be fabbing up a mount for them(which is honestly about the extent of my fab skills) - but I have something atypical in mind...
Stampie
SuperDork
5/28/17 10:16 a.m.
At the least make the feed to your fuel pump master switch switched power with the key on.
In reply to Stampie:
That was my initial plan, but I discovered the power to the FP relay is hot all the time. I could track down a circuit somewhere, probably, that is keyed, but that leads me down the rabbit hole towards ripping it all out and starting over - which quite honestly is what I'd like to do, and possibly will someday(LS swap would be a perfect time). For now, I just want to make it easier to live with. The FP switch under the hood proved it will work(although the FP seems to be running normally now regardless), but certainly isn't very convenient.