No, you have to normalize it. That is fixed in the projection postulate. You might check it in the book on "Quantum Computation and Quantum Information" by Nielsen and Chuang or here: https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics#Density_matrix_formulation

It depends what you want to calculate. Assume you make a measurement and obtain outcome 'a'. The effect of this measurement is given by sandwiching the density matrix by measurement operators (projective or otherwise). We call this sandwich 'r'. The state after the measurement is obtained by normalizing 'r'. The trace of 'r' gives you the probability that 'a' was measured. Thus, depending on what you want to calculate, you might be interested in the trace of the unnormalized density matrix 'r' (which is lost after normalization). However, physical states are always described by a density matrix with trace equal to one.

The relevant formulas can be found in the link posted above.

Because all the operators in question are linear, it might be beneficial to make calculation first and then normalize the answer. 

If you want to obtain a proper wavefunction (density matrix) in order to calculate operator averages you indeed need to renormalize  it.