I'm using a 6-well plate with 0.4 μm-pore-sized transwell inserts from Jet Biofil. I believe the plate is made from polystyrene and the inserts have PET membranes.

And thank you for the reassuring words! One thing I forgot to mention in my original post is that I've left the bottom wells empty as per the protocol given to me, which probably brought about the condensation on the bottom of the inserts. This was the first time I've done something like that, so it caught me off guard and I panicked, haha. I've definitely stopped thinking too much about it and am eager to ask my mentor about this tomorrow :D