Yeah that's fine. Every tantalum capacitor (the big black/orange ones) has one end connected to voltage and one end to GND. You can connect to just the GND end without issue. He probably did it that way so the solder joints would hold the all-in-one PCB against the Wii securely.
You probably...
As kowabungus said, it would be wise to remove your cells and check their current voltage. That will tell if it's the PMS under-reporting or a cell fault.
Photos of the cell and PMS wiring would also be useful
Genuine ones and/or tips from comparable name brands are better than the cheapo ones, but not by a huge margin. The tip is just a bit of metal heated by the element in the iron handle. The major differences are longevity and how snugly the tip fits against the element.
For hobby usage either is...