I have the official answer from Apple itself: my lid wouldn’t stay closed because the battery had swollen, causing the case to bend. (No, unfortunately my Early 2015 13-inch MBP is not a part of an exchange program by Apple. Oh well.) The more the battery swole, the worse the flex/bend. This not only prevented the lid from closing properly, but also explains why there could still be keyboard marks on the screen even though the lid wasn’t able to fully close.
This quote sums up my experience exactly:
It may start with a trackpad that just doesn’t click like it used to. Or you may notice that your laptop doesn’t sit flush to the desk anymore. Eventually, your laptop’s case may start to warp and bulge, and you’ll know for sure that you’re dealing with a swollen battery. While a swollen battery can be dangerous, we’ll show you how to deal with one so that it becomes nothing more than a minor inconvenience.
The repair cost $100 for labor and $180 for parts, so $280 plus tax. They replaced the entire case including keyboard and trackpad, though they did leave the bottom plate intact, which was disappointing because I had installed some vinyl pads after the original feet wore through. Not sure if I have any different hardware than before other than those two things, but they were authorized to use refurbished parts if needed. When I asked what that meant, the team member said that these are parts that may have been pulled from other machines, but tested and confirmed working properly. Gave them my macbook on early Friday afternoon, got it back early Monday afternoon.
Knowing that the swollen battery’s out of the way, now even more interested to upgrade that hard drive given this may be the best 13-inch macbook ever built. (And now re-built, again, after it’s already had one keyboard replacement.)
I’ll never buy another Apple product again without AppleCare. It used to not be necessary, but these days it’s a requirement.