Is it common practice to discuss your employment status with your boss
prior to starting the process of buying a house?
I've never been asked that question. I've never asked that question. And I've never heard of that question being asked by a prospective house buyer. So in my experience, it is not at all common practice.
Basically, should I ask my boss, "Hey, your not planning on downsizing
in the near future, are you?"
In some public companies, your boss could not disclose that information if a downsizing event were upcoming. And even in private companies, few employers would want to provide any sort of solid assurance, as it could potentially be held against them should circumstances change.
Basically, you need to trust your instinct on what you see happening at the company, your knowledge about its stability and past experience. Since you say that the company is "is stable and growing" it seems as if you already have your answer.
Remember that "in the near future" is short term, while a mortgage is generally long term. You need to be financially prepared no matter what happens.