To answer the first question, if we are using the churban to motivate us to do teshuvah, we need to be mis’abel on what we lost so that we desperately want it back, and will do what we need to in order to get it back. This is not to disagree with what others have said that aveilus is itself the avodah, but to point out that they work together.
To answer the second question (which I had thought of yesterday as well when we said Av Harachamim), the status and dinim of moed are nidcheh to today as well.