1. Whose job is it? Our priests have relegated this to lay persons.
2. If Lay persons are allowed to do this where may I find the directive that allows this?
3. I have seen lay persons casually going in and out of the Tabernacle. Is there a directive or instruction that allows this?

1) If you mean setting up the chalice etc., that's been the job of lay sacristans for ages.
2) So it doesn't need to be explicitly said. It's customary.
3) I hope it's not casually. Oftentimes sacristans will ask me to check the amount of hosts in the tabernacle, but they can check on their own for a good reason.

I've recently found myself coordinating and giving guidance to altar servers, (mostly college Military cadets) and suddenly realized the priest wants the Chalice set up for him, it is something I have never done and frankly am not comfortable with. I spoke with another priest I know, he said he has never relegated that responsibility to anyone, so he never dealt with the issue and was not aware of how to direct me to the guidance. Perhaps it is as you say, we just starting doing it a long time ago, so what begins as an abuse, becomes customary then it becomes an accepted norm, like altar girls.

It's normal, if there is a procession, for the chalice and paten to be set up on the altar before Mass. There is no reason that a sacristan cannot do this. If you're scrupulous about touching the sacred vessels then maybe wear some white cotton gloves?

Here's a picture of St. Therese preparing the sacred vessels:

Is the outer Chalice veil and burse optional I have never seen our priests using these items?

I don't know enough about the new rite of Mass to venture a guess

To chime in, Sacristan has done this for every Mass I've ever been even near.

Veil and burse are optional.