The hot start problem is a well known issue and mainly is a result of the low fuel pressure the D-Jet injection system uses (29psig). You can increase this pressure but then the fuel mixture controls need adjustment. For idle mixture, the adjustment is on the ECU and for load mixture, on the Manifold Pressure Sensor. But to adjust these, you need a means of measuring %CO in exhaust (or AFR). You can't do it blind or by feel. I installed a permanent wideband O2 sensor with AFR gauge on my own car. Portable units are available.
To overcome overheating, I would first look at timing. The NA specs call for less advance than the Euro cars. This for emission reasons. Try setting the idle timing at about 5deg BTDC. Then rev engine to 3000rpm and check how far it advances. It should get to about 27-30BTDC. If not, turn distributor more counterclockwise. If you don't see that sort of advance, the internal centrifugal weights in the dizzy could be sticking. Running retarded causes overheating.
The hot start problem is not likely due to vapor locking in fuel rails. It is most likely caused by fuel flashing to a mix of vapor and liquid as it passes through injectors. This reduces the injector capacity so cylinder only gets a portion of what ECU is trying to provide. As a result mixture is way too lean, rpms drop and engine cuts out. Keeping tank full and running fuel pump several times by switching key on and off can help get some cooler fuel into rails. A switch to bypass the fuel pump relay would do the same thing. But this relies on fuel in tank and supply line being cooler. Not always the case when parked on hot tarmac!
I tried all sort of "fixes" including some of those mentioned for the hot start problem. But in the end, more advanced timing, 32-34psig fuel pressure and mixture adjustment resulted in no further problems.
There is a section here on the 107 forum that has a lot of info on the Djet system. It is in Encyclopedia Germanica in 107 forum stickies under D-Jetronic.
The D-Jet is a unique system. It can work well, but often not understood.
Have fun with your new project!