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I don't know if 1 and 4 can be done, but 2 and 3 can.

See the documentation on

• Power Series Rings, Power Series, Power Series Methods,
• Multivariate Power Series Rings, Multivariate Power Series.

For instance

sage: R.<x,y> = PowerSeriesRing(QQ); R
Multivariate Power Series Ring in x, y over Rational Field
sage: h = (exp(x) - 1)*(1/(1-4*y))
sage: H = h(x,exp(x*y)*y)

If you feel ready to write your own software, why not instead write a class in Sage for multivariate power series with coefficients expressed as symbolic expressions in the exponents? You could then use all the simplification tools already in Sage.

I don't know if 1 and 4 can be done, but 2 and 3 can.

See the documentation on

• on Power Series Rings, Power Series, Power Series Methods,
• , Multivariate Power Series Rings, Multivariate Power Series.

For instance

sage: R.<x,y> = PowerSeriesRing(QQ); R
Multivariate Power Series Ring in x, y over Rational Field
sage: h = (exp(x) - 1)*(1/(1-4*y))
sage: H = h(x,exp(x*y)*y)

If you feel ready to write your own software, why not instead write a class in Sage for multivariate power series with coefficients expressed as symbolic expressions in the exponents? You could then use all the simplification tools already in Sage.