Hi Eleanor!
I see you're confused with one of the most confusing points in learning Farsi. but don't worry, it's easy.
there is no difference between می کنم and میکنم. in my experience, the first one is more common, though some people, like me, prefer to connect می and کنم, the second one.
As I told you in the previous line, there is also no difference between خوشحال and خوش حال as the first one is way more common. I can say I've never seen someone writes خوش حال. But a short distance between خوش and حال doesn't change the meaning.
the story of the word گرسنه is totally different. As you see, there is a ـه at the end of the word. there are many words with the same ending such as نامه، خانه، شانه، خامه، کاناپه، .here the ـه is pronounced "e" not "h" and no letter must be added to it. so in گرسنه ای we use an additional letter just like "ا" here to refuse pronouncing it like "h". that means if there's no "ا" the word will be گرسنهی and as you know this form, ـهـ is always pronounced "h". it will be " gorosnehi" which is wrong.
PS: there's a couple of exceptions for ـه. for example in some words like مِه and بِه you read it "meh" and "beh". it means "h" is completely pronounced. ( just for more information :) )