1. The girl is aged 13.

2. The girl is 13.
3. The girl is 13 years old.
4. The girl is at the age of 13.
5. the girl is at age 13.

(Are they all grammatical and the same? Is #1 correct?)

#s 2 and 3 are fine. The rest are highly unusual and not used.

Maybe 2 or 3 but 4 is okay too

All of the sentences you provided are grammatical, but there are some subtle differences in meaning and style.

1. "The girl is aged 13." - This sentence is grammatically correct, but it is less commonly used in everyday conversation or writing. The word "aged" is not typically used to describe someone's age; instead, we simply say "the girl is 13."

2. "The girl is 13." - This is a straightforward and commonly used sentence to state the girl's age. It is clear and concise.

3. "The girl is 13 years old." - This sentence is also common and completely correct. Including "years old" provides specific information about how the age is measured.

4. "The girl is at the age of 13." - This sentence is grammatically correct, but it sounds slightly formal or old-fashioned. It is more common to say "the girl is 13" or "the girl is 13 years old" in modern usage.

5. "The girl is at age 13." - While this sentence is grammatically correct, it is less commonly used than the previous options. "At age 13" can be seen as more technical or formal, and it is more common to simply say "the girl is 13."

Overall, options 2 and 3 are the most commonly used and preferred ways to express the girl's age. Option 1 is less common but still grammatically correct, while options 4 and 5 are less commonly used but still grammatically valid.