Exaple sentences
No. It should be "Is the grammar right in this sentence?"
The sentence "Is Mary not Happy" is correct grammatically as a question. It defines Mary as the subject and not happy as the object. In the other sentence, though grammatically correct, it could be confusing. Not Mary could be construed as all people but Mary. I would use the first sentence, as it is clearest.
They are both happy. NOT They is both happy.
It's an adjective.
I am happy to show you that. When I am happy, I show it.
Happy is an adjective.
'Birthdays (no hyphen) to come' is just fine: We wish you many happy, healthy birthdays to come.
I was happy about being invited to my friend's birthday. THA AND
Why is it that you are not happy to do your own home work
You were very happy yesterday
I think that you would write: Hi Julie! Happy New Year!
I was happy and exited. there :)