(too long for a comment.)
should be true by looking at the character table of $GL(2,p)$ in characteristic 0, see for example the book "A Journey through representation theory" by Gruson and Serganova on page 147.
seems to be true, but I do not know a reference (maybe books by Gordon James?). Here is a way to test it for a given prime with MAGMA (you can input it in http://magma.maths.usyd.edu.au/calc/ )
p:=7; G:=GeneralLinearGroup(2, GF(p)); SIMS := AbsolutelyIrreducibleModules(G,GF(p)); temp_field_sizes:=[]; for i in SIMS do Append(~temp_field_sizes,#BaseRing(i)); end for; MaX := Maximum(temp_field_sizes); MaX;
when the result is p again, then the prime field is a splitting field.