(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.

(too long for a comment.)

1. The character table of GL(2,p) in characteristic 0 can be found for example the book "A Journey through representation theory" by Gruson and Serganova on page 147.

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/ ). The code also works for any other finite group to get a minimal splitting field over prime fields.

   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. Magma confirms that the prime field is a splitting field for all primes <20.

(too long for a comment.)

1. 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.

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.

(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.