The answer to Q1 is ``No." 
is
Proof. is true if and only if the mirror of 
is
not printable. But the mirror of is itself. Hence is
true iff it's not printable. By the definition of the biconditional, we thus
have two possibilities: viz., either is true and not printable, or
is printable and false. The second possibility violates one of the assumptions
of the puzzle (viz., that 
never prints sentences that aren't
true). So we are left with the other disjunct: is true but not
printable. QED
As to 
, it must be false (since it's negation is
true). Therefore by the hypotheses governing the puzzle this sentence is not
printable. If we reinterpret `printable' as `provable' we can conclude that
is, as we say, undecidable, that is, neither it nor its
negation is provable.