The completed sentences with the appropriate articles are:
Sheldon is an honest man. (An is used before words that start with a vowel sound.)
King Arthur was a just king. (A is used before words that start with a consonant sound.)
A lion let go of a mouse. (A is used for indefinite nouns.)
I have fixed an appointment with the doctor. (An is used before "appointment" because it's the first mention, and the is used before "doctor" because it's likely referring to a specific doctor you already have an appointment with.)
Will you take a picture of us? (A is used for indefinite nouns.)
I broke the vase my brother brought. (The is used before "vase" because it's likely referring to a specific vase that was previously mentioned.)
My mother has a polka-dotted umbrella. (A is used for indefinite nouns.)
The team organised a friendly match with the Presidents of both the countries. (A is used before "friendly match" because it's the first mention, the is used before "Presidents" because it refers to a specific group, and the is used before "countries" because it refers to specific countries that were previously mentioned.)
We had a mulberry tree in the garden. (A is used for indefinite nouns.)
The Eiffel Tower lights up at night. (The is used before "Eiffel Tower" because it's a specific landmark.)