If you are writing for a formal or academic purpose, you must write out twenty first century.
yes
She may not have WRITTEN it, but she is the singer of 21st Century Girl.
The most intractable challenges of the 21st century.
because she wanted to
The ancient Egyptians far suppassed the intelligence of the 21st century Egyptians.
Yes, "Century" should be capitalized when used in reference to a specific century, such as the 21st Century.
21st century.
We are in the 21st Century.
It is in the 21st century.
the 21st century
~this is the 21st century
Yes they do have batteries in the 21st century and had in the 20th century.