Yes, that is correct.
I was tired but happy and a little wistful when we finally arrived home.
My little brother was very tired, so I gave him a piggyback ride.
Only if there is no semicolon between "tired" and "you".
Yes, it could be, for example in the sentence 'I am tired of all this arguing, as are you.'
No, it is not a complete sentence.
Yes
Yes, that is correct.
I was tired but happy and a little wistful when we finally arrived home.
No, it should be little more only.
My little brother was very tired, so I gave him a piggyback ride.
I'm starting my year with this cute little puppy - is the correct sentence
This sentence is a reported sentence and 'that' is a conjunction. It introduces the 'that clause' - he was tired.
Only if there is no semicolon between "tired" and "you".
Yes, it could be, for example in the sentence 'I am tired of all this arguing, as are you.'
"We were informed it will be corrected today" is a correct sentence. However, it might sound a little better to say "We were informed that it will be corrected today."