How do the Chinese celebrate the coming of a baby? And do the Chinese have "baby showers"?

Chinese don't have baby showers and no celebrations until the baby's born.

Once the baby's born, the family will prepare "red eggs" (basically eggs cooked in red dye so that the shell is red) and give out these eggs to friends and family. Family will also cook a big pot of "ginger vinegar" and give some them to friends and family.

Then depending on the family traditions, some families will have a banquet when the baby's 1 month old (some have it on the 100th day). This is the day when the family officially introduces the baby to friends and family. Before that, the baby and new mom is not supposed to be going out visiting people.

Normally the Chinese celebrate "Man Yue" (literally A Full Month), that is, they mostly celebrate the fact that their baby has reached the following month.

Besides the obvious auspiscious associations with the cycle of the Moon in the Chinese Lunar Calender, I suppose (this I'm not so sure) in the old days, having a baby born is no big deal (hello, large families, largest population) but the fact that a baby has survived beyond the first days is cause for greater celebration.