Are kernel-level services considered as a part of the kernel?
That can be debated, but for me the answer is yes.
You might run into a purist who considers only the code from the manufacturer in the kernel to be part of the kernel and extensions as an add on the kernel. I would ask you instructor for his answer, you are the customer, right?
If so , does writing kernel-mode services mean extending Operating System?
For me it surely does.
The easiest way to think of this is adding a device that requires installing a kernel mode device driver. Once the kernel mode device driver is installed all the code has access to the device driver in the same way all the code has access to the kernel. That is simplified but you should get the analogy.