No, probably not, it is more than 20 inches long and most underseat storage does not allow more than 18 inches. I recommend a smaller bag for under the seat.
I worked in a bike shop for almost 6 years and you said it right. Its pronounced "Tooley" so yes, you got it right. Just like "Yakima" is pronounced like YAK (like the animal) KEE - MAH
In Swedish (Thule is a Sweden brand) it would be pronounced more like Tu+Le with th “u” in “tu” being similar to “ue” in “nue” and “le” similar to french “les” or like “le” in “lego”. The exact “u”-sound doesn’t exist in english though.