you can look up TID/SID combos in pandora's box in rng reporter. it's possible to find out if TID and SID are a matched pair and also what PIDs can be shiny for a given TID/SID. also, looking up shiny vs. non-shiny eggs is the exact same process in rng reporter's time finder. the only difference is that for shiny eggs you plug in your TID/SID. as for pid, it determines most everything about a caught poke and just about everything fro a hatched poke. for example, look at your PID.(in decimal) if the last digit is even, your have ability 1. if the last digit is odd, you have ability 2. IVs and their link to PID are determined on a method 1 poke due to this: 4 rng calls are made for a method 1 poke. the first 2 calls determine the PID, the second 2 determine its IVs. if the first 2 calls are the same(in that you're getting the same PID more than once), the second 2 calls have to be the same. so, if call wwww and xxxx happen, each time they happen the last 2 calls, yyyy and zzzz, will always be the same. AR changes certain chosen values, but not all of the other values linked to the changed ones.
hatched stuff is different becasue the PID/nature/ability, etc is determined by one RNG(IRNG - mersenne twister variant) when the egg is spawned with the daycare man and the IVs are determined by a different rng(LCRNG - linear congruential RNG - the "normal" rng for this game- used by most things) when you pick the egg up from the daycare man. different rngs determine each part of the hatched poke at different times.
sync checks are done because in dppt/hgss some legitimate pid/iv combos are skipped and not used at all. pokemon you encounter due to in-game events(sinjoh, eevee, roaming pokes, etc) can use any of the 4.2 billion possible PID-IV sets available. (PIDs are 32-bit, so no more than 4.2 billion- 26 billion are possible, though not usable without longer PIDs) standing legenardies and caught stuff don't have all of those available to them. some are skipped when those pokes are generated, so it isn't possibnle to find them on wild stuff. something would be hacked if, for example, one of the skipped PID-IV sets appeared on a wild poke.