I have a used Dell Inspiron 5558 laptop, which didn't come with an OEM
power supply. I picked up a 65W "universal" type for it off Amazon, and
it's showing some weird behavior.

First problem is that it often refuses to charge the battery when
connected to line power. Maybe 1 out of 10 times I've plugged it in the
battery charges, the other times the charging LED remains off and the
charging indicator on the screen remains at 0%.

It sometimes has difficulty starting up from line power at my home as
well; you have to hit the power button a couple times to get it to power
on instead of coming up for a few seconds then immediately shutting
down. However, it seems to always work fine off the local Starbuck's
wall power. Still doesn't charge properly, though.

The BIOS reports some incidences of "Unknown AC adapter detected!" but
not all the time.

Before I run out and make sure to pick up an OEM supply, is there any
way to ensure this is certainly a problem with the adapter and not the PC?

Maybe a BIOS update is required?