No crank, no start

About a month ago, my car wouldn't start. No crank, no start. I got a new battery for it because it needed it anyways and had the starter rebuilt with new brushes. Solenoid is the same. That started the car and for about 2 weeks, it started every time. And then one time, it just didn't start. It won't jump start although it will push start.

I ended up replacing the Interlock Relay, adding much needed oil (2quarts) and cleaned the plugs, but put the same ones back in because I don't have any money right now. Now, it starts intermittently.

When the car is cold, it will start eventually. When I turn the key, I hear a click and then some weird noise that sounds like something is dying in my engine and if I keep doing that, eventually, it will crank and start like it's supposed to. When it does start, it's as if there's not a problem at all. Cranks and starts like it always did.

Once the engine is at normal running temperature, I guess, the RPMs drop a bit and I get this weird knocking noise coming from the engine. And the whole car shakes a little bit. Possibly, the knocking sounds like it could be coming from a sensor, but I'm not sure.

I suspect possibly one problem might be the ignition switch, but even that, we put a jump wire from the battery hot to the back of the ignition and it does work, but not every time. If I'm bypassing the ignition altogether, shouldn't it start every single time?

Also, someone messed with the idle screw and now when it does start, I sometimes get smoke at first. A fair amount of smoke, I think. Please help me. I don't know what else to try.

Thanks.