For whole Cherry Tomatoes? No. They don't provide consistent temperature readings when canning so the Food Preservation authorities consider them unsafe to process whole. They should only be used like regular tomatoes, sliced or chopped.
As for your tomatoes, I would definitely keep them refrigerated.
If you are a beginning to canning, I strongly suggest you buy the Ball Blue Book of Preserving, which is the bible for preserving food. Cost is less than $10.
Support bacteria. It's the only culture some people have.