Why is my Mac able to write to my NTFS hard drive?
Over the network it does not matter what file system is used on the hardware drive, because the computer hosting the drive (making it available through the network) will access it.  Your Mac will just make generalized requests over the network for files, folders, etc. and the host will access the file system and serve the files. Network requests are standardized and the host answers them and converts between those and the actual file system on the hardware. 

What you were reading about not being able to write to NTFS, was people who either had a partition with NTFS on their native HD in the Mac computer or who were attaching USB drives.  In both these cases, the Mac has to access (and possibly modify) the file system on drive directly (without an intermediate agent) and needs to be able to deal with the specifics of the file system directly.

And no, there is no danger of damage (well, not unless you or the Mac intentionally delete files or move folders that are needed otherwise).