Any specific type and brand of oil to prevent oil burning (00 corolla with worn piston rings)

From your description I'm betting the valve seals are shot. Why? Letting off the gas when the car is moving increases vacuum which sux oil passed the seals.
Does hcl cleaner do good?

should I be using conventional oil or synthetic? My main goal is to prevent the car from throwing a rod through the block or something. I’m fine with putting in the oil but I can’t have this engine destroy itself. Will synthetic stop gunk up from happening? Should I be using a lighter (5w-20) or heavier weight oil (10w-40)? I feel that the heavier weight oil will slow down oil consumption but oil won’t be getting to necessary parts of the engine. What do you think?

The car doesn’t have a cat anymore

Also if there is a number I can call you at. I’d rather do that then text lol. Up to you though. If so, direct message me ur number
Just get in touch with @High Performance Lubricants here on BITOG. They are the one making the HPL Engine Cleaner product, and likely will answer your questions far better than I will.
We use esters in our Engine Cleaner that should ultimately clean up the rings and pistons. The challenge in your situation is the consumption. For the best results you would add the cleaner at 1:5 ratio with the oil. The oil you are using is fine. As you add makeup oil you would need to make sure you top off the engine cleaner at the same rate to maintain the ester concentration.

An additional thought after looking at the pistons in the picture above would be to run a PEA based fuel system cleaner in your tank. I would probably do that continually for a couple thousand miles. Avoid the “Italian tune up” as EGT’s over 500f will kill the PEA.

This would be a slow and safe method for you to follow.