The main takeaway from your post is first and foremost that your carpet is damp. The case is a good second line of defense, but you really need to find the source of the infiltration (for multiple reasons)... and I too would suspect the drains. Cleaning is not difficult, and even if your not mechanically savvy, it's not something that you need to have the dealer do, HOWEVER...you stated that you paid them to, and they did not; if service was billed and not done, they really should do so. Additionally, if they are cleaning the drains they should be able to provide information on the condition of the pans.
Regarding the battery, I always disconnect as part of the "better safe than sorry" approach.
Welcome to the forum.
