My DA has been parked for the last three years. Finally getting around to diagnosing the no-start condition. I have power with the key turned on, dash lights up, but it only starts with the key 10% of the time. Most of the time I will turn the key to start it and just get a click. The car will run fine if I pop start it.
I suspected the starter and replaced it, but that didn't make a difference. Now I'm thinking it could be my ignition switch. So my questions are:

1) Has anyone else experienced this, and what was the issue?
2) How can I test my ignition switch so I'm not throwing away money by guessing and replacing parts?
3) If the ignition switch is bad, can I replace it without having to change the lock cylinder and key?

Any other suggestions on diagnosing this would be appreciated.

*Additional info:
91 Integra with JDM B18B swap. Still running factory OBD-0 ECM.
Funny electrical issues include dash lights randomly going out while driving, front corner turn signals flashing together when using signal (both left and right), radio intermittently cuts out while driving.
I think these are separate issues from the no-start, but I figured giving as much info as possible is best.

 

It starts when you push start it because the engine is being forced to turn over and start.
Wont start with the key 90% of the time because the ign. switch is bad.
2 screws hold the switch onto the key cylinder and a few clips.
No need to replace key cylinder.
If you have a new starter and it still wont start 90% of the time, its the ign. switch.

The rest of the wiring sounds like a nightmare.
Turn signals sounds like an open in both left and right wires and when either signal is used, both signals go to work.

Dash lights may be loose connecter at the back of the cluster or loose wire on the connector.

Radio might also be a loose connector or wire on the connector.

 

Like beating a dead horse, I'd check the body to battery ground. Made a huge difference in my car. The blinker issue is common for these cars. Take your blinkers out, and cut the green wire to the blinker that's acting weird (on both my cars it was the upper blinker) then just run new wire down to the green wire in the bumper blinker. Also make sure u have a full charge on your battery and double check your wires are tight on the starter. If it's the ignition switch, the car should run if u hold it in start (obviously don't do it for long! )