1. Please take a picture of the dog that is running.

2. Please take a picture of the dog
running.

(Is #2 correct? In Sentence 2 'that is' is missing.)

Both are correct.

Both sentences are fine. In #2, "running" seems to be used as a present participle modifying "dog."

Thank you. What about the following?

3. Please take a picture of the running dog.
(Is it grammatical?)

Yes. That's a great sentence.

Yes, sentence 2 is correct. In this case, the phrase "that is" is not necessary because the verb "running" functions as an attributive verb and modifies the noun "dog." This construction is commonly used to describe an action or state of the subject. So, by saying "Please take a picture of the dog running," we understand that you want a picture of the dog while it is in the act of running.