"What do you do to honor your children's cultural and ethnic backgrounds?"
In as many ways as possible. We talk to our child about his heritage from both sides of the family. We show him pictures of his elders and tell him stories about them. We talk to him in English as well as other languages we speak. When he is old enough, we will expose him to the various cuisines, as well as family recipes. We will encourage him to study history and to honor his ancestors.
Around the house, we put up a mix of decorations that tell about our heritage. That's as much for our child as for us, his parents. It's for the whole family. We also do it to honor our departed loved ones.
It doesn't matter if you are from mixed stock. In this age, people travel, move and inter-marry so much that it's really common. All you have to do is know what your cultural background(s) are and learn to appreciate them all. That does not mean you disregard American values. If you live in a country whether as a citizen, immigrant, expat or refugee, you should honor that country, its laws and its heritage.
"Do you think there is a time of their life when this is more or less important?"
Yes. It's important! This needs no explanation. You can see it in families where culture and tradition are preserved. But again, taking pride in your heritage should not the same is trampling on American heritage (or wherever a person may live in).
I don't know about "heritage months." What would be better is to research holidays in your culture and what they mean. Then celebrate it with your family in a way that is meaningful to you.
We will encourage our kids to travel when they are older so they could see for themselves where their parents and elders from!
I won't comment on the other stuff because I don't think your question is about that.
Good luck!