Less gravity when moon is up?

Not anything of significance. Do the math. On the surface with a radius of 1737 km, the moons gravity is .16 times that of earth. Now the distance from us to the moon is 363100 km. so we do (1737/363104) squared and then multiply that by .16. That means the moons pull on us is 3.6 times 10 ^ -6 time the earth's... or about 3 millionths the pull of earth.

Gravity isn't lessened, it's just that the direction of gravity 's pull on things is altered slightly (That doesn't answer your question at all, does it?). You still get pulled down by the earth when the moon is directly overhead, but you also get pulled up a veryveryveryveryveryveryvery little bit.

Let's say "Yes," and use tides as an example - the moon pulls water towards it, thus we get tides. There is a very slight difference in gravitational pull. But keep in mind that you don't notice tides even in good-sized lakes - only in inland seas and the great lakes, so the use of 1g = 9.8 m/s^2 is still accurate enough for government work.

NASA, though, uses slightly different figures (I'm sure).

yes