Yes, you are correct.
First, "centering" is meaningless in a moving window as long as everything is "centered" it all works out. The point is that you can use centered values but it won't change anything since it's all relative.
Second, you will have negative values at the start of the data set. There are various ways to overcome this such as zero-extending the data set in the negative range, periodically extending the data, mirroring the data, using a forward looking window instead of a backwards or centered window(but this can cause problems).
Your problem is analogous and overlaps with taking the derivative of a function of a finite interval.
Let $f(x) = x$ only on the interval $[0,1]$
what is the derivative at $f(0)$ and $f(1)$? Well, it doesn't exist. But instead we could either continuously extrapolate $f(x)$ to some larger interval then figure out what $f(0)$ and $f(1)$ or use the left and right derivatives.
You'll have to figure out how you want to deal with these end cases.
For example, if you used a forward looking window through out the data set you'll then end up with issues when you reach the end. You could use a forward window for the first half of the data set then a backwards window for the last half but then you'll have issues in the middle.
You could use a gliding window which starts in a forward looking position and transitions to a backwards position(and it will be a centered window at about the center of the data set).
Another method is to use a window that expands and contracts. It starts off as a size of 1 for the first data point, 2 for the second, 3 for the third... until it reaches it's maximum... then near the end of the it shrinks.
If you "discontinue" the loop you'll throw away a lot of data for no reason. It's a simple method and works when the data set is large and the window is small. But in some cases you have a window comparable to your data set. In your example, you'll essentially be throwing away the first 80 data points(well, you will have no computation for those data points).