In standard written English dialog, unless the word following the closing quotation marks is a proper noun, it would be in lower case (not capitalized).
For example: "Stop! Thief!" shouted Sally.
"Stop! Thief!" Sally shouted.
In your examples, "Ring! Ring!" The telephone rang.
would be correct if the writer was making two separate statements. The first statement describes the sound being made, the second describes the action making the sound.
"Ring! Ring!" the telephone rang.
is technically correct if the writer is describing the ring of the telephone bell as if the phone is speaking - "Ring! Ring!" said the telephone.
It's the same as saying "Woof! Woof!" the dog barked. This gives us the impression that the dog is "speaking", or is the center of the action.
Compare that to:
"Woof! Woof!" The dog barked. This describes a situation in which a dog in the vicinity barked, and someone who was nearby heard it. The dog's bark is a sound effect, not an actual line of dialog.
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