There is general convolution algebra type formalism that you can try and use, see Chriss and Ginzburg's "Complex geometry and representation theory". Chapter 8 in particular is very much in line with the your "source of examples".

In general this can be hard. Heck, consider even the case that your local system is trivial and your IC-sheaf is the (shifted) constant sheaf on $X$. Then you are asking to compute the cohomology of the space. This may be a non-trivial endeavor depending on your space.

An ideal example where things work out very nicely is that of flag varieties and the IC-sheaves are those corresponding to Schubert subvarieties. Then these Ext-computations can be carried out combinatorially in the Hecke algebra. Soergel's papers on this and related topics are particularly enlightening. A related point here is that in this situation considering hypercohomology as a functor to graded modules for the cohomology algebra of the flag variety is full and faithful. This result also generalizes to projective varieties with $\mathbb{C}^*$-actions. This is a result of Ginzburg "Perverse sheaves and $\mathbb{C}^*$-actions". Similar ideas are also worked out in some of Springer's papers on spherical varieties. Related are also the moment graph techniques that can be found in papers of Braden, MacPherson, etc.

Regarding, "replace cohomology with equivariant cohomology". I am assuming you want to compute $Ext$ in the equivariant derived category. Then similar techniques as above can be tried. In the presence of suitable assumptions, often the equivariant calculation reduces to the non-equivariant one due to formality. Instead of trying to flesh this out let me just refer to Soergel's "Langlands philosophy and Koszul duality". A number of examples are worked out in there.

In general though, there is no magic pill.

There is general convolution algebra type formalism that you can try and use, see Chriss and Ginzburg's "Complex geometry and representation theory". Chapter 8 in particular is very much in line with the your "source of examples".

In general this can be hard. Heck, consider even the case that your local system is trivial and your IC-sheaf is the (shifted) constant sheaf on $X$. Then you are asking to compute the cohomology of the space. This may be a non-trivial endeavor depending on your space.

An ideal example where things work out very nicely is that of flag varieties and the IC-sheaves are those corresponding to Schubert subvarieties. Then these Ext-computations can be carried out combinatorially in the Hecke algebra. Soergel's papers on this and related topics are particularly enlightening. A related point here is that in this situation considering hypercohomology as a functor to graded modules for the cohomology algebra of the flag variety is full and faithful. This result also generalizes to projective varieties with $\mathbb{C}^*$-actions. This is a result of Ginzburg "Perverse sheaves and $\mathbb{C}^*$-actions". Similar ideas are also worked out in some of Springer's papers on spherical varieties. Related are also the moment graph techniques that can be found in papers of Braden, MacPherson, etc.

Regarding, "replace cohomology with equivariant cohomology". I am assuming you want to compute $Ext$ in the equivariant derived category. Then similar techniques as above can be tried. In the presence of suitable assumptions, often the equivariant calculation reduces to the non-equivariant one due to formality. Instead of trying to flesh this out let me just refer to Soergel's "Langlands philosophy and Koszul duality". A number of examples are worked out in there.

In general though, there is no magic pill that I know of.