Need help figuring something out in Immunology

Let’s say there’s an extracellular peptide like a bacterial toxin for example >>> gets phagocytosed by APCs (B cells, dendritic cells and macrophages) >>> Ag undergoes processing via the exogenous pathway and is presented on the APC surface along with an MHCII molecule.

Now a dendritic cell (with this particular MHCII/processed Ag on its surface) presents this to a naive CD4 T-cell >>>and with the costimulatory B7/CD28 and LFA-3/ICAM-1 interactions ultimately ends up activating the same CD4 T-cell >>> IL-2 secretion and clonal proliferation. (Question: is this activated T-cell specific for the Ag?)
CD8 T-cells don’t read MHCII and hence are not activated at this point in time.

This activated CD4 T-cell now interacts with a B-cell >>> costimulatory CD40/CD40L binding >>> secretes cytokines which stimulate class-switching and Ab production.
But these CD4+ cells could be Th1 (gamma-IFN and activates macrophages) or Th2 (IL-4/5/6/10/13 and class switching to IgE for parasites and IgG/A for other stuff). If the peptide is extracellular (as I mentioned in the beginning) do both Th1 and Th2 get activated with the Th2 response predominating?

Is my understanding of this thing correct? Also the CD40-CD40 ligand interaction occurs between what and what? [any random B cell vs. B cell that presents that specific Ag] and [Th1 versus Th2 CD4 cell, or both]? Finally what happens first - does the Ag presentation influence Th1 vs Th2 or does pre-differentiation of Th into Th1 and Th2 dictate the subsequent immune response depending on the type of Ag?

Similarly if there’s an intracellular peptide, it would be processed endogenously (TAP of the ABC family) and presented with MHCI by
1) Any nucleated cell to a CD8 cell – leading to cytotoxic killing of the nucleated cell.
2) MHCI on a dendritic cell to a naïve CD8Tcell >>> same B7/CD28 etc interactions and again killing of the dendritic cell (?)
3) The B cells are nucleated and have both MHCI and II, but since an intracellular Ag is processed endogenously, it is never presented with an MHCII – and hence no Th-B cell interaction (CD40/CD40L thing)
4) Since the default Th differentiation pathway is Th1-directed, in response to an intracellular Ag, there is thus no Th2 response and thus no B cell class switching.

Is HbsAg or HbeAg for example, an intracellular or extracellular peptide? Because if it was intracellular, what I’ve written above can’t account for anti-Hbs IgG or IgM. Where am I wrong?