Some questions about MM proposition

I have some questions after reading MM theory and I cannot find answer for myself.
First, MM theory and also static trade off theory seem to find an optimal capital structure to maximize firm value, however I wonder whether shareholder’s value is also maximized at the same time as firm value? (because Equity value = Firm value - Debt value, hence firm value may increase but the debt may increase faster). The goal of management after all is to maximize the shareholder value, not the firm value, isn’t it?

Secondly, in curriculum, they illustrated MM proposition with tax by an example in which the firm issues debt to repurchase exactly amount of equity, then VL= VU + tD. I wonder what happen if the debt is still issued but not to repurchase equity (the number of outstanding stocks remains the same). Then, does the equation VL = VU + tD still hold? (assume EBIT is the same before and after debt issuance)