For B and T cell activation is this correct:
An APC presents antigen on MHCII which stimulates helper T cell.
B cell phagocytosis a pathogen and presents MHCII to helper T cell which gives rise to plasma and memory B cells.
T cell is stimulated by APC with MCHI and does not need helper T to stimulate it.

Thank you.

I recommend reviewing page 161 in Classroom Companion 2 for a great illustration of the two systems.
Starting with the 1st antigen exposure. An antigen presenting cell (or macrophage for most MCAT passages) will phagocytose an invading antigen, stimulating helper T-cells. Helper T-cells will stimulate B and T cells as well as differentiating into memory helper T cells to stimulate B and T cells if a second exposure of the same antigen were to happen.
B cells will be activated by both free antigens and helper T cells. B cells will then differentiate to memory B cells (to prepare for a second exposure) and plasma cells (they secrete antibodies to target the antigen in question).
T cells will be activated by helper T cells and infected cells that display the invading antigen. The T cells will differentiate to either memory T cells (to prepare for a second exposure) or active cytotoxic T cells that attack infected cells.
Regarding your second post, I'm not quite sure what B-cell receptors refer to: antibodies are created by plasma cells that target antigens corresponding to the invading pathogen.
Hope this helps,
Thank you