Very early in my ownership (driving it home) my car illuminated both the oil pressure light and Ex gas sensor light. Long story short, the oil pressure warning light switch was bad; removing the boot on the switch showed oil oozing out. Installing a replacement did the trick.
If you suspect the cold start injector of activating when it is not supposed to and dumping excessive fuel into a hot engine and causing hard starting, it's easy to eliminate that possibility.
1. You can just unplug the cold start injector's electrical lead and if that makes a difference, then your issue is the thermo-time switch, which is what activates the cold start injector.
2. If unplugging the electrical lead does not make a difference, try a pair of pinch-off pliers or equivalent to pinch off the fuel supply hose going to the cold start injector. If that does make a difference yet #1 did not, then the issue is the cold start injector not properly sealing itself off, in which case you'd need a new cold start injector.
Leaking injectors dumping fuel into an engine can lead to thinned out oil which can lead to loss of compression. Does your engine oil smell gassy or look thin? Smoking gun!
A fuel pressure reading is always helpful in these cases. An easy but dangerous way to connect a temporary gauge is the feed hose to the cold start injector. It's dangerous because the Cold start injector fuel line nipple is old and brittle and identical cold start injectors are crazy expensive if you break that nipple off while trying to remove the hose. Depending upon what your fuel pressure gauge kit has for fittings, it might be better to T into the fuel hose right after the pump or right after the main fuel filter.
During hot no start events, check the electronic ignition box located in the right engine compartment scuttle. These are generally warm to the touch, but if yours is really hot, that might be an indicator of an ignition issue creating the hot no start.
Hopefully a worn engine is not your ultimate issue, but if you suspect low compression as the hot no start culprit, compression tests that compare cold and hot values will tell you everything you need to know.
Chain parts stores have loaner programs for common test equipment like fuel pressure gauges and compression testers. While you have the fuel pressure gauge kit, check to see if it is also a vacuum gauge. Knowing how to read a vacuum gauge is a lost art but those who know what they are looking for attest that they are valuable tools, even for engines that won't start.