Galerkin method is used for discretisning the differential operator in space for the boundary-value (PDEs) and initial-boundary-value problems (time dependent PDEs). While it may be used for pure initial value problems, I don't think it is worth the effort.
You will be better off by using the conventional time integration techniques such as Newmark-beta, generalised-alpha or Runge-Kutta schemes by transforming the equations into the space.
Evolutionary problems involving parabolic and hyperbolic equations in any number of space dimensions are formulated in such a way that Galerkin methods, using a local basis, can be used to find approximate solutions. These methods are particularly suited to problems involving general regions and derivative boundary conditions.
https://academic.oup.com/imamat/article-abstract/7/2/241/706654?redirectedFrom=PDF
Time integration schemes can be applied. Galerkin methods are also applicable.
Yes it can be used. You may wish to present the problem explicitly. Some problems may be more easily solved using numerical time integration schemes.
YES. GALERKIN METHOD CAN BE USED TO SOLVE SUCH PROBLEMS. HOWEVER THERE ARE OTHER SOLUTION METHODS.
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