Did I calculate the expected value and the variance correctly?
He who formulated the exercise is wrong, and this is why I am posting a full answer to a homework question.
This is a classic example where manipulating recursive relations can lead to different representations that may appear "different" and with different properties.  

$$\Delta Y_t \equiv Y_{t}-Y_{t-1} =  \mu + Y_{t-1} + \varepsilon_{t} - \mu - Y_{t-2} - \varepsilon_{t-1}$$

$$\implies \Delta Y_t = \Delta Y_{t-1} - \Delta \varepsilon_{t} \tag{1}$$

At the same time 

$$Y_t = \mu + Y_{t-1} + \varepsilon _t \implies \Delta Y_t = \mu +\varepsilon _t \tag{2}$$

The right hand side of Eqs. $(1)$ and $(2)$ represent the same process. 

In any case, neither of these are $MA(1)$.

Moving forward, which one to choose?  

A soft adoption of Occam's razor indicates that eq. $(2)$ is the simplest. A bit more specifically, we note that the manipulation resulting in $(1)$ hasn't "saved" us from the existence of a unit root and non-stationarity in a clear way. 

Both therefore suggest to adopt $(2)$, that says that  $\{\Delta Y_t\}_t$ is the sum of constant and a $WN$ process.