This is kind of a weird one. Let me break down my findings:
The cold checks from what I can see are the checks it does when you first turn the ignition on. It will send a pulse to the bulbs for a few seconds, at which point it determines if a bulb is dead or on its way out. The LEDs will flicker for a few seconds as they receive the pulses from the LCM during the test. The reason the incandescent bulbs do not flicker is because they require significantly more time to illuminate. The pulses are so short that the incandescent bulbs don't have enough time to illuminate with those short pulses.
Coding out the cold check will eliminate this flickering. The downside is that you will not get an immediate notification that a bulb is out. You'll only get that notification once you go to try to use the bulb. So, say your brake light bulb is out but the cold check is disabled. Your car will not tell you a bulb is out until you press the brake pedal. If cold checks were on, it would tell you right off the bat after turning the ignition on due to the test pulses checking for it.
Warm checks I'm not 100% sure about. My theory is that the warm checks are pulses sent at set intervals while driving. A few people say it's around 5 minute intervals that the LCM sends the warm pulse checks. Based on research I've done and experiences I've had with my car with the warm checks enabled, that information is backed up by what I'm seeing. I've had warm checks enabled but cold checks disabled and I've had people following me tell me that my lights flicker every few minutes, so that makes sense.
I've recently disabled all warm checks with my car in order to completely eliminate flickering. The downside to this is that it seems that it doesn't notify you of failed bulbs at all this way. I tested this while waiting for laundry at the laundromat after seeing your question on the forum. I disabled warm and cold checks, then I took out my turn signal and brake light bulbs.
With the turn signal bulb removed, the turn signal stays constant and does not hyperflash. It also does NOT tell you that a bulb is out, which is a bummer because you wouldn't have a clue based on its normal flash rate. The brake light, however, did tell me about the bulb out as soon as I went to press the brake. I'm not sure why the turn signal failed to misbehave when it should have but the brake light notification acted as expected.
I'd like to re-enable the warm checks and watch the car for about 15 minutes to see if the flickering does indeed happen. If so, that will prove my theory. Ultimately, there was a time when cars didn't have checks for burnt out bulbs, so personally I'd rather just do a visual inspection of my car from time to time and have super bright, good-looking LEDs that don't flicker.
Yes, I do have a sedan and they are different from the 2-missing-door variants. I bought bulbs from eBay, meaning there is no "brand/model" I can give you. The best I can offer you is "25W CREE CANBUS 1156 Amber" or "Red". I have the amber for turns and the red for the brake light portions. The bulbs between the amber and red look exactly the same physically, and have (Qty) 5, 5W CREE XP-E chips (for a total of 25W), which are super bright even in direct sunlight. The red is brighter than the amber due to the wavelength of the light (or so I've been told), but the amber is plenty bright.
I did recently order (3) pairs of 25W Red bulbs for my running lights (bottom row in the tail lights, and trunk light), but I found that they are WAY too bright when in parking light mode and are blinding at night. I don't know why they don't run at half brightness like they are supposed to, but I'm sure there's a technical reason that I don't understand, and frankly don't want to put the effort in researching it. As a result, I reverted the running lights to incandescent bulbs, which is probably a good thing anyway since they are way cheaper to replace and they are on ALL the time (I have them as DRLs).
I'll take a pic of the LED bulb next to the factory 1156 incandescent bulb to give you an idea. I'm not sure what socket your front bulbs are though, so the factory length of yours may be different than mine.