Unless I'm misunderstand your question, since you specified "+3 unlock", once you have the relevant ABCD Sheet, you can enter the basement area of that sector at any time on any job. The sheet doesnt need to be re-farmed.
As far as easy goals, and I'm presuming you're alluding to ones that you can get case/saph/star:
Casket #A1 - Vanquish 5 Abject foes.
Casket #A2 - /heal in the area between Gate #A1 and the Abject Leeches.
Coffer #A - Vanquish Abject Obdella.
Casket #B2 - Open any Locked Gate #B.(J-8), (K-8), (M-8)
Coffer #C - Vanquish Cachaemic Bhoot within 5 min of spawn (depends on luck and your kill speed)
Casket #F1 - Interact with the Diaphanous Bitzer while visibly wearing or lockstyling 5/5 pieces of Empyrean Armor of any upgrade level (including the base piece) for your current job.
Casket #G1 - Stand still within 6 yalms of the Diaphanous Bitzer while continuously targeting it for 30 seconds.
Casket #H1 - Leave and re-enter Sector H.
If you have the means to safely tackle D fomor with your job and trusts, D1/2/Coffer should be doable. I'd save these goals for last though.