Is the market maker for a particular stock also the market maker for that stock's options?

Most anybody can buy or sell options, and a big or unusual order could certainly affect the underlying stock. But it's the new information about the order that does this, not the market maker. An efficient market digests new information and adjusts prices accordingly.

The answer to that is "it depends". Let's start with "what is a market maker"? In some exchanges the market is made by a specialist responsible for a stock - and the exchange does not do options. On other exchanges it is a tag assigned to someone behaving in a certain way, and he or his company can do what they want - they may or may not do options or they may only do options, THEIR decisions (and yes, I know some people doing only options market making for stocks).

The answer generally can not be given because the environment is extremely diverse AND - the world and all the different exchanges makes it even more so.