320mm is an odd distance for centres. What is on the other side of the wall, or is it a load bearing stud wall? (This is normally evidenced by double thickness sole plates/headers, but not always). I'd be concerned about the studwork at the bottom left of the first picture - the horizontal trimmer terminates before it reaches a vertical stud, leaving the end unsupported. It needs to be doubled up and tied into the vertical stud to the fight of the back box with a vertical piece to the floor at the other end if it doesn't already have one
TBH that wall looks to be a plasterboarded, but is it plaster skimmed on the other side? If so if you should definitely use screws to add your trimmers, not nails, as nailing risks cracking the plasterboard joints. Yes, you ideally need to install two horizontal trimmers at the top, but the PB should hold stuff together for you whilst you cut out. You will need two vertical trimming pieces to carry the PB at the sides of the cutout, but unless the weight of the vanity us excessive, a single horizontal trimmer ar the bottom should suffice
BTW forget the joist hangers, though, they are completely inappropriate and unnecessary for this type of work. Set you work lines with a laser level or a long spirit level and just use 5 x 90 or 5 x 100mm screws - that's all you need