This is a community wiki answer which serves as a collection of responses from Techidiot, Maria, and Wesley.
Maria solved most of the word search, determining that:
The pet is at the vet. See her answer for a story using the words found within the word search.
Techidiot solved the sudoku one of the 100 some ways that it can be solved, however it was not the intent. In chat he made his way into figuring out:
The sudoku has the numbers 12, 3, 6, 9 in the N E S W positions, resembling a clock. (With a little help from Wesley here too)
Wesley finished up the puzzle, figuring out the meaning behind the note:
His answer includes a clock explaination, and that the numbers were interpreted as time zones. The abbreviations lead to CAT, EAT and NUT.
He then noted that you were supposed to mix EAT to get ATE. This left you with the final note of The CAT ATE NUTs.
Nuts are in fact toxic to most cats and dogs when ingested, making it reasonable that they visited the vet.