Submerging in ice water will help, but it's the sides of each plate that's touching ice water and the top/bottom. Many times that is just touching the hose water. More worth it to me to have the chiller mounted to my stand than floating in a bucket, but I can see why you might.
For cleaning, you want to flush it backwards and forward, but you have to use a cleaner. Recirculating with a pump is best, but even a good soak with a cleaner will help a lot. Star San is not a cleaner. A proper bake at the right time and temp will kill everything in there, but best to not have to kill a bunch of mold each time. Trying to sanitize with Star San with any kind of debris in there will never fully sanitized.