Step 1: Understand the question.
We must pick the two correct statements about recombinant DNA technology. So we look for the statements that are true.
Step 2: Check statement A.
A says selecting recombinants by inactivating antibiotic resistance is cumbersome. In practice, antibiotic selection is a standard and handy method, not a clumsy one. So A is not the best choice.
Step 3: Check statement C.
C says any protein-coding gene expressed in a foreign (heterologous) host gives a recombinant protein. This is a clear textbook fact, so C is correct.
Step 4: Check statement D.
D says the sequence GAATTC is expected once in about 4096 nucleotides. A six-base site appears on average once every $4^{6}=4096$ bases, which is the standard figure. This statement is treated as correct. So D is correct.
Step 5: Compare with statement B.
B about chimeric DNA is loosely worded and less precise than C and D, so it is not chosen. The two firm, standard facts are C and D.
Step 6: Pick the correct set.
The correct statements are C and D.
\[ \boxed{\text{C and D}} \]