Autocorrelation of a Shifted Sequence
Hi: I think there are two different vewpoints to your question so I'll mention both. 

First Viewpoint: ( Time Domain )

What you have written for the covariance is not wrong but you need to understand what you're calculating. Those numbers you are calculating are unscaled covariances of an MA(n) process  assuming A) that the variance of the noise term in the  MA(n) model, $\sigma^2$ = $1$ and B) the coefficients of the MA(n) model ( i.e: the $\theta$ ) are all 1.   So, the covariance of the shifted MA(n)series does stay the same because the 5 numbers ( A) the variance and B) the 2 autocovariances at lags 1 and lag 2 and lag -1 and lag -2 which are happily symmetric ) are staying the same. You get the other zeros because, for some reason, when you shifted the sequence in the time domain, some extra zeros arose. I'm not sure how that happened but it shouldn't. atleast not in the time domain. 

Second viewppoint ( Frequency domain )

I'm just learning DSP so can't say a lot here but, from a time series frequency domain perspective,  if $y$ has  covariance generating $g_{y}(z)$, then that $X(z)$ filter can be viewed as an MA filter of order(n) so that if the MA(n) filter is applied to $y(t)$, resulting in say, $X(t)$, then the covariance generating function of $X(t) = B(z)B(z^{-1}) g_{y}(z)$ where $g_{y}(z)$ is the autocovariance generating function of $y$. 

All of this quite nicely covered in Hamilton's "Time Series Analysis" if you have that book. I highly recommend it for time-series but it's mostly time domain material with frequency domain coverage somewhat scattered. Still, book is really great.