Only if they are in two different clauses, since the form of to be with I is "am".
Examples:
"I can see that it is raining outside."
"I am the one who is responsible."
(unless we use the facile "The letter I is after the letter H.")
"They both have great humor" is a sentence.
You can use the word "both" to refer to two things, people, or groups that are being mentioned. For example, "Both Sarah and John attended the meeting."
They are both happy. NOT They is both happy.
It is both a question AND a sentence.
both sides
Yes
Hammer and nails are both the subject of that sentence.
It is a sentence because it contains both a subject and a verb.
we both went outside
Both of the tests were easy, to me.
both and
Yes both words have same meaning!