PKHeX fixes common errors when setting to the save file. You can either check legality from the box or disable the automatic fixing under the set to save option.
Would require knowing which species is being battled as well as what level.
I can't see this being added to PKHeX (maybe as a helper function to Core and used by a separate app), just bump the level by a few manually and start using it when it's underleveled.
Use the batch editor for arbitrary randomization, refer to the FAQ for usage.
Randomization macros in the main editor are only useful for randomizing numbers, you can always scroll randomly to pick a random choice from the drop-down.
Just added the Ability Checks.
https://github.com/kwsch/PKHeX/commit/a4e04fc2fe445bcc7edceb95641fada0d4a3f1fd
As for the held items, PKHeX is nice to the user and doesn't remove items on transfer across generation if the item still exists in the later generation. Saves the hassle of reattaching items.
PID is just a number that is used in some generations to derive Nature / Gender / Ability / Form (gen3 only). It's just a random 32 bit value, any number is valid.
PID-IV combinations are another story; they vary per generation and origin console and only matter for gen3/4. PKHeX does not do any PID-IV relationship when picking a PID; it only generates for Nature/Gender/Ability/Form. Adding in IVs and a RNG method would result in many impossible IV setups... hence it's up to the user to manually pick a PID for their IVs if that's something they really care about.