Let $(\Omega,\mathcal{F},\mathbb{F},\mathbb{\mu})$ be a filtered probability space.
Market efficiency implies that the stock price process is Markov with
$\mathbb{E}[f(X_t)|\mathbb{F}_s] = g(X_s)$ for $0 \leq s \leq t$
where $f$ and $g$ are Borel measurable functions.
It additionally implies that the discounted stock price process is a martingale w.r.t. probability measure $\mathbb{\mu}$ and filtration $\mathbb{F}$ with
$\mathbb{E}^{\mathbb{\mu}}[X_t^*|\mathbb{F}_s] = X_s^*$ for $0 \leq s \leq t$
While the discounted stock price process is a martingale the stock price process itself should be a submartingale w.r.t. probability measure $\mathbb{\mu}$ and filtration $\mathbb{F}$ with
$\mathbb{E}^{\mathbb{\mu}}[X_t|\mathbb{F}_s] \geq X_s$ for $0 \leq s \leq t$
I agree with the others Markov does not imply martingale and vice versa.
There are many sources on empirical tests for these properties.
In my opinion these assumptions are not unreasonable. Markov property only says that all past information about the stock price process (historical prices, historical volume, etc.) is incorporated in the current price and therefore only the current price is relevant. I believe that it is logical to assume that publicly available historical information is already priced in. For instance even the anomalies violating weak form efficiency (e.g. the January effect) tend to disappear over time as market participants trade on the information thereby incorporating the information into the price. Assuming that the stock price process is a submartingale only says that in expectation the future stock price should be greater than or equal to today's price. Intuitively, investors would not participate (long positions) in the stock market if prices were expected to decline. Take stock price process
$$
dX = \alpha Xdt + \sigma XdW
$$
A submartingale implies $\alpha \geq 0$
For most assets I don't believe that that is an unreasonable assumption.