Let $({W}_{t}^{(i)})_{0 \leq t \leq T}$, $i=1,2, \dots, n$ be a sequence of independent standard Brownian motions on the probability space $(\Omega, \mathcal{F}, \mathbb{P} )$ and let $(\mathcal{F}_t)_{0\leq t \leq T}$ be the associated filtration. In addition, let $(\theta_{t}^{(i)})_{0\leq t \leq T}$, $i=1,2,\dots,n$ be adapted processes and consider \begin{align*} Z_t = \exp \left( -\int^t_0 \sum^n_{i=1} \theta_{u}^{(i)}d {W}_{u}^{(i)} - \frac{1}{2} \int^t_0 \sum^n_{i=1} (\theta_{u}^{(i)})^2 du \right). \end{align*}
How to show that $Z_t$ is a martingale under the measure $\mathbb{P}$.
Your equality (**) holds if $\theta^{(1)},\cdots,\theta^{(n)}$ are independent.
About your last inequality, how should we understand them? Should we understand that it holds in this specific case? Then of course it is true, and those are even equalities (if you have the above mentioned independence). Else, should we understand that for random variables $X_1,\cdots,X_n$, $$ \mathbb E\left[\prod_{i=1}^nX_i\right]\le\prod_{i=1}^n\mathbb E[X_i]? $$
The latter is obviously false. Because that would imply for instance by taking $n=2$ and $X_1=X_2=X$ that $\mathbb E[X^2]\le\mathbb E[X]^2$, which is false for any non-constant random variable $X$.