Are Maxim lithium ion battery chargers and fuel gauges compatible with lithium polymer batteries?

All Maxim parts that are compatible with li-ion batteries are also compatible with lithium polymer batteries. Lithium polymer batteries are a subset of li-ion batteries and have an almost identical chemistry. The main difference is the packaging, which is usually a prismatic or pouch shape wrapped in foil instead of hard-cased cyliners. They are electrically interchangeable with the same charging profile and voltage curves, so they can be used for the same applications.

Why is the A9 N.C. pin connected to BYP on the MAX77818EVSYS?

The NC pins (i.e. A1, A9, C4, C6, E2, E4, E6, F3, F4, G2, H1) are floating and not internally connected. Page 23 of the MAX77818 datasheet states, "N/C pins are left floating and can be connected at board-level if needed." The MAX77818EVSYS connects the A9 N.C. pin to BYP because you must go through the A9 bump to get to A8 bump on the top layer.