GTOs of higher quantum numbers?
Has a Gaussian expansion of STOs of higher quantum numbers (for example, 2s or 3p STOs) been performed


I am unaware of such fitting. Typically, basis sets using higher angular momenta (cc-pV$n$Z, def2-$n$ZVP, pc-$n$) are fitted on total energies or correlation energies. See https://www.basissetexchange.org/ for examples and references.


  Are there any GTO integration schemes for Cartesian GTOs of higher principal quantum numbers?


Yes, there are. I have seen the following approach in action before: 


hand-crafted algorithms are used for common, low-angular-momentum cases 
more general schemes, but still optimized schemes for higher cases 
a catch-all, not very efficient, unspecific algorithm above a certain threshold