Unless I messed up the notation, it seems that Note 5.3 of this paper of Medina, Noll, Rowland on Iterated primitives of logarithmic powers answers Question 2.

It seems quite likely that the OP is well aware of the above paper, given this recent paper (2011) (Amdeberhan, Koutschan, Moll, Rowland, The iterated integrals of $\ln(1 + x^n)$)?

Unless I messed up the notation, it seems that Note 5.3 of this paper

Luis A. Medina, Victor H. Moll, and Eric S. Rowland, Iterated primitives of logarithmic powers, International Journal of Number Theory 7 (2011) pp 623–634, doi:10.1142/S179304211100423X, arXiv:0911.1325 (pdf)

answers Question 2.

It seems quite likely that the OP is well aware of the above paper, given this recent paper:

Tewodros Amdeberhan, Christoph Koutschan, Victor H. Moll, Eric S. Rowland, The iterated integrals of $\ln(1 + x^n)$, International Journal of Number Theory 8 (2012) pp 71–94 doi:10.1142/S1793042112500042, arXiv:1012.3429 (.ps file).