Starting problems
 

Will not start by cranking. Cranks slowly but will not start.
With one guy pushing on flat ground it will start.

Without much to go on, it sounds like a weak battery or poor battery connections (positive or ground). Slow cranking probably also means low voltage to the ECU and ignition. That might be why it will start while pushing - the starter is not requiring power, and so the voltage to the ECU/ignition stays high enough for it to start.

I had same problem with a 2nd gen RX7 (88 turbo). Would not start with starter, only by push starting. Starter sounded fine but would not start.
Was intermittent, Has car for 4 years, only did it about 5 times. Took to Mazda dealer, no help.

The faster a rotary engine cranks the more compression it makes. If you are cranking REALLY slow you'll have problems starting.

DJSeto has some videos in his build thread, you can listen to how it sounds cranking with the original and with the upgraded 2kw RX-8 starter -

https://www.rx7club.com/build-thread.../#post12522640

If it's substantially slower than that you will have problems. This is like any other car, you need a good battery, good electrical connections,and a good starter.

Dale