When you’re discussing “rates” or “prices” or “costs” you can ask any of the questions you mentioned, which have slightly different meanings:
What is the room rate?
This asks exactly what it sounds like- it’s asking for the room rate. You’d expect an answer like “The rate is $100 per night”.
How much is the room rate?
In this context “how much” is essentially the same question as “what”. You’re asking about the cost of something, which “what” is adequate for. “How much” can be asked about any sort of measurable quantity. “The rate is $100 per night” is also the type of answer you’d expect here.
But, you can also use “how much” without a word like “rate” because it’s implied you’re asking about the cost:
How much is the room?
Like before, you’d expect an answer that tells you how much it costs. There is no way “how much” can refer to anything besides cost in this context.
How is the room rate?
This is the one that is slightly different. You’re obviously asking about cost, but with this question you might want to know how it compares to other room rates, or how “good” or “bad” it is. An answer to this could be “It’s pretty good compared to other hotels in the area. The rate is only $100 per night.” Or, “It’s way too expensive*- don’t stay there.” In other words, the question is about the cost, but doesn’t require the actual cost to be part of the answer.
*edit: maybe “high” is a better word than “expensive” for describing a “rate”, but you get the idea.