Back in the late 70's my uncle had a Suzuki jeep type thing we ran around in (slowly).  IIRC it had a 360 cc motor, canvas top,  High / low range stick, military style tires and perhaps the starter was also the generator?  Just curious as I am getting into this Suzuki thing.
Tio Pick

Either an LJ10 or LJ20 - was it air cooled or water cooled?

http://www.lj10.com/

I would guess water cooled but it has been a long time since then and I could be wrong.  He had a "lake pipe" installed so he could reach down from the driver's seat and remove / replace a (radiator) cap on the pipe.  We never challenged it so to speak but if you could walk it, you could drive it in southern Michigan farm country.

In my mind a "lake pipe" is more exhaust related than cooling system, so that would be no evidence of water cooling.

true, lake pipe(s) are an exhaust thing that you can remove a cover and have open exhaust.  Uncle Willy's was just fabricated to use a radiator cap for convenience.  Did not necessarily improve anything except the loudness / fun factor.  I realize that most small engines expect a certain back pressure for proper operation.  BTW, I have not read the Texas Traffic Code in many many years but mufflers were only required inside city limits back in the day.  Speed limits between towns were rather loosely enforced as well..  Stretches in excess of 30 miles..  we used to do one 40+ miles stretch through the King ranch without dropping below 100 mph ...  Obviously not in a little Zuki.  Again, Zuki was in southern Michigan farm country / Irish Hills

Did it also only have 1 door across the front ?

Smile to only one door across the front.  Uncle Willie did have one of those stashed away in a barn as well but this was a Suzuki jeep I am trying to ID