Is there any way to remove the home screen "numbers", i.e. (red rectangle):
and instead have one more row of app shortcuts?
I am using a Samsung Galaxy S5 SM-G900A with Android 4.4.2.
This is not possible on Samsung's Stock Launcher but many alternative launchers have the possibility to display even more rows by shrinking them in height.
Try a third-party launcher as nearly all Samsung stock applications tend to be unoptimized for any Samsung device; especially in memory consumption.
E.g. using Nova Launcher Prime you can remove the scroll indicator as follows:
Result (the number of rows and columns of the desktop grid can be changed as you want):