(too long for a comment.)
2. 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.