[Zapmeister]

disclaimer: i haven't done this stuff in years.

1.

PV = nRT is the relevant equation. so:

the only thing you're changing is T, and the thing that's going to change is V. P,n,R are all constant here.

so: V2/V1 = T2/T1 = 573/303
since the height you started with is proportional to volume, have h2/h1 = 573/303
so h2 = 573/303*10cm = 18.9cm
so h2-h1 = 8.9cm.

i'm pretty sure all the other numbers in the question are irrelevant.

2.

um. i don't think this is doable unless there's extra information in the diagram that you didn't tell us. your equation "(2atm)V3 = (.036945)(T3)" is correct, there's no way to explicitly get V3 and T3 from here (like what's to stop the answer from being T3=T1 and V3=V1?)

to get V1,V2,P2 you can use PV = nRT like you're trying to do for V3, which i guess you've already done.

3.

pressure difference = density * g * height difference

539 N/m^2 = density * 9.8 N/kg * 0.05m
density = 539/(9.8*0.05) kg/m^3 = 1100 kg/m^3 which suggests your answer would have been right except that you left out g in the formula.