Speaking of duct tape, I'm using it as a bandaid right now. It really is a universal 'fix all.' haha

Yes. The rules of DIY: If it moves when it shouldn't, use duct tape[0]. If it doesn't move when it should, use WD40

[0] A slightly heretical alternative here is to use cable ties, daisy-chained if necessary for larger jobs.

Arguably cable ties are the plastic edition of pre-tape bailing wire

High temperature silicon based sealants are used for sealing engines. Hardware stores usually sell small tubes (50ml or so) which is enough for odd repair jobs like that.

As you can see in the photo, there is discoloration on the plastic. Yes these have almost melted several times.

Correct, but it doesn't touch the coffee nor do I have to touch that part of the lid, so I'm not too worried (also I just pretend my buddy is getting old and blind with age)