Cumpleaños uses the most common form of Spanish compound word. They are constructed of the third person singular indicative of a verb and the plural form of a noun. So you have
Cumple+años cumpleaños complete years = birthday
Lava+platos lavaplatos wash dishes = diswasher
Abre+latas abrelatas open cans = can opener
As a rule these nouns are all masculine and don't change between singular and plural.
Here is a link with some more examples
I definitely hear mi. But the difference in sound of me and mi at normal Spanish speaking speeds is so slight. Part of what helps you understand is that your brain knows what should be where. That's why you will have people say "I thought you said [some random unrelated or strange thing]. That which you know you should know eventually will smooth out the minor variations in pronunciation that occur naturally every day. It's a function your brain does by itself eventually, but doesn't work as well of course without context which is Duo's weakness. But you do need your mind to self correct your ears a lot.