"I am busy in the morning."
Native here. 早 doesn't mean morning. It means "sometime relatively early" and is a very general word. It's actually much more used frequently as a character (word building block) than as a word. In the rare case it's use as a word itsef, it's likely when you casually greet a friend: ”早！" (similar to "Morning!") Back to the main topic: ”早” in ”早上" is not a word itself, just a building block. "上" doesn't have much real meaning and is just a syllable filler, as Chinese words tend to be two characters long (mostly). Or you can say it's simmilar to another example ”早间" as in ”早间新闻" (morning news). Here "间" is also a filler but more close to having a meaning "during". At a later stage you will appreciate more on how flexible Chinese characters are being used to build actual everyday vocabulary as compared to being used as is. When using multiple characters to form a word, the meaning of the characters don't always align with part of the meaning of the resulting word. It's usually more abstract and metaphoric. It's a fascinating topic, but also one of the unique charasterics of the Chinese language. Hope it helps, be patient and have fun!
的 often gets dropped when speaking. So something like 我的妈妈 will become 我妈妈
Because then your father would say "Nihao mang, Wo shi ni de baba".
We translate 是 to 'am' in English, but that does not mean we can always translate am to 是. 是 can be though more like an equal sign in math, rather than be used to describe something. In a conversation, you could probably get away with not saying 很, but it would feel odd so to make it flow better 很 is used.