I have the following 3 load cases with some springs:

L1: W Restrained weight case (HGR case)
L2: W+T1+P HGR case
L3: W+P1+H SUS case.

When I check displacements in L3 at spring locations, it is not zero. Why is it so?

Ceasar's default and what is perfered by most users most of the time is for the springs to carry the sustained support load when the system is hot. In the sustained weight case, your load is your Hot Load plus or minus the hot dispalcement times the spring constant.

After the adding and subtracting is done, your springs are carrying a different load than if your system was hard supported. This Load is called the Cold Load. Since this load is different than the exact sustained load you will see some deflection at the springs.

Go to Caesar Spring Help. Caesar provides a lot of options for you to size your spring just they way you want them.

That means it uses the cold load in L3

Yes

Actually, it sets the spring at the 'Theoretical Cold Load' for all cases and the actual spring load in all the other cases change with the deflection based on the spring constant.