First, thank you very much for the detailed list of symptoms - I always appreciate when a person asking for helps in turn helps those of us trying to solve a mystery. Secondly, thank you for posing such an interesting (though for you frustrating) problem. Believe it or not I have actually seen something similar to this before. To summarize:
The problem occurs randomly as far as gear combo and timing, but has perhaps somewhat of a rhythmic component.
It's a prolonged sound, similar to ratcheting.
It happened after a derailleur adjustment.
Changing gears sometimes affects it.
It may occur/start more when pedaling down on the right side
The sound seems to be coming from the right side.
Given the above there is one thing I can think of that is the most likely candidate, and that is interaction between the pulleys and the chain, with cog/chain being the less likely candidate.
- Pulleys are designed to have some play in them, must turn smoothly, and have plastic teeth that can be damaged fairly easily.
- The derailleur or pulley assembly as well as the derailleur hanger can sometimes be knocked out of alignment.
- The chain has only a minor amount of tension when it comes off the bottom of the cassette, and chains sometimes have or develop a waviness from side to side or a twist.
- Most people have more strength in their right leg so that frame flex, which can affect drive train alignment, is more likely when pedaling down on the right side.
I believe what is happening is some of the factors above combine at the right time to unship the chain from the bottom pulley so that it is running on the cage. It would make a ratchet sound as the links pass roughly over the cage instead smoothly turning the pulley, and of course the sound would be on the right side of the bike. It would not be as loud as the chain mis-indexing on a cog. It also would not be felt, and the chain not engaging the cogs properly would tend to be.
The challenge becomes determining if that is indeed what is happening, and then how to fix it. If what I propose is indeed occurring then the bottom section of chain would tend to shake back and forth when the sound happens - but it would be hazardous to try to see that while riding!
Given your lack of experience I would strongly suggest you take the bike to a shop. If you start to change things it could make diagnosis more difficult or introduce more problem.
First have them check for be grease or scratches on the bottom of the pulley cage. Then ask the mechanic can put the bike in a stand and pedal while intentionally flexing the pulley cage out of line, in order to force the chain to unship from the pulley, and see if that is the sound you have been hearing (it will sound louder than you experience when on the bike). If it is then you're much closer to resolution.
The next step is correcting the problem. Have the mechanic confirm that the derailleur hanger and the derailleur/pulleys are properly aligned. If they are not, then correcting that could be enough to fix the problem. It's important someone with the right tool and a good eye do the work, as a bent derailleur or pulley assembly is not always easy to spot. Of course one could just try a derailleur swap, but given the intermittent nature of the problem the mechanic may not be able to determine a difference.
Next would be looking at the chain from the rear as the cranks are turned backwards to see if it "snakes" from side to side or has twists in it, as that can contribute to the problem. Tight links will show up as well, because the pulley assembly will "jump" as the bad section(s) pass through it.
The pulleys should also be checked to make sure they do not have excessive play, rotate freely, and do not have damaged teeth, and the cage should not be bent in toward the pulleys.
The mechanic would have to determine what needs to be done to correct the problem depending on what is found. Feel free to show the mechanic these notes, rather than trying to remember and explain everything. A good wrench will either know/understand already what I've laid out or will be open to trying out my hypothesis.
p.s. Just realized you did not say why you were changing the adjustment screws in the first place - was something occurring to cause you to try adjusting them?