A
Once you get $\displaystyle dM_3(t)=H(t) dB(t)$, you are done proving that $\displaystyle M_3$ is a local martingale, whatever the right-continuous adapted process $\displaystyle H(t)$ is. You don't have to specify that $\displaystyle H(t)$ is a martingale or some kind of product of martingales.
As for proving that $\displaystyle M_3,M_4$ are true martingales, you have the following result: if $\displaystyle M$ is a local martingale with $\displaystyle M_0=0$, the following is equivalent :
(i) $\displaystyle M$ is a true square-integrable martingale ($\displaystyle E[M_t^2]<\infty$)
(ii) for all $\displaystyle t>0$, $\displaystyle E[\langle M\rangle_t]<\infty$.
Remember that $\displaystyle E[\langle \int_0^\cdot H(s)dB(s)\rangle_t]=\int_0^t E[H(s)^2] ds$ so that you can easily check this criterion in your case.