The two circled LED's "flicker", but after a few minutes, they stop flickering and stay lit. Why? I would like them to stay flickering.

 

The two circled LED's "flicker", but after a few minutes, theyView attachment 270885 stop flickering and stay lit. Why? I would like them to stay flickering.

Do you have the 100uF series capacitor there on purpose? The output is now AC coupled and there exists a DC path through both LEDs to ground.

Place a 10nF capacitor from Control to GND instead.

The Control pin sets the threshold of an internal comparator. Having it at 6V could cause problems as the timing is no longer standardized.

 

The 555 circuit "alternates" lighting the LEDs (yellow then red then yellow then red over and over) without flickering.

My neighbor has 5 lanterns on a post in his front yard. Each lantern has orange flickering LEDs in it and they look like flickering flames inside.
Today I bought a solar garden light at The Dollar Store for $4.00CAN and it has 12 orange LEDs in two vertical rows, an IC, a battery and a solar panel. It also looks like a flickering flame inside because the IC lights each LED at random. The lantern charges the battery in sunshine with the LEDs turned off then turns on the Flickering LEDs when it gets dark at night.

 

Do you have the 100uF series capacitor there on purpose? The output is now AC coupled and there exists a DC path through both LEDs to ground.

Place a 10nF capacitor from Control to GND instead.

The Control pin sets the threshold of an internal comparator. Having it at 6V could cause problems as the timing is no longer standardized.

View attachment 270888

The two circled LED's alternate on and off. The 100uF causes them to not fully go off, but rather they get dimmer and brighter. It works really well for about a minute, then they both stop and are on, (no flashing).

 

The 555 for some reason stops working and the output gets clamped to Hi or Lo. In either case, the 100 uF capacitor no longer plays any role. Both the led.s run off 9 volts with 440 Ohm resistor in series.
If the 555 is overheating, it will shut down.

 

The 555 for some reason stops working and the output gets clamped to Hi or Lo. In either case, the 100 uF capacitor no longer plays any role. Both the led.s run off 9 volts with 440 Ohm resistor in series.
If the 555 is overheating, it will shut down.

If it is an overheating problem, any ideas how to solve this?

 

If it is an overheating problem, any ideas how to solve this?

Forget my second statement. It will not apply here.
The 100 uF capacitor will cause the voltage at pin 3 to be almost double the supply voltage. (18 vs 9). Do you need that 100 uF capacitor?
If the clock rate of the 555 is Low, the 1 LED will Flash bright and then slowly dim (as the capacitor charges/discharges) with the 2nd LED becoming from Dark to Dim. And this Flash-Dim / Dark-Dim will alternate between 2 LED's.
If there is no capacitor, it will be a simple flash between the 2 LED's.

 

The 100uF causes the led's to have a 'flicker' effect rather than flash all on and then all off.

I can't figure out how you are getting 18V.

Side note...I breadboarded this again today and it seems to be working great?????

 

The tendency for plastic breadboards to provide erratic results is legendary. Soldering irons are easy to use and highly recommended.