There is no such selection on the X4000 as it is not designed to power 4-ohm speakers, whereas the selection does exist on the flagship 4520CI as it is designed to power 4-ohm speakers. The E/X models can generally power 4-ohm speakers to at least to moderate TV listening levels without the AVR going into protection mode, perhaps higher with very efficient speakers.
After you are finished using a Network source, try pressing/holding the ENTER button to STOP it rather than simply changing to another source.
Try doing a soft reset by unplugging the TV and the AVR for 10 minutes. If still no joy, you may want to consider disabling Anynet+ on the TV and instead connect an optical cable from the TV to the AVR.
On the XX13 models, there is no dedicated HDMI Pass Through setting as there is on the newer E/X models, however, a work around if you're passing the cable/sat box is to connect it directly to the TV with HDMI and optical from it to the AVR.
The Denon X2000 is a great lower mid level model that can likely satisfy all of your requirements. Review post #2 of the Denon E/X Owner's thread to learn more about the different features of each of the models.
Congrats on your new purchase. When the additional features of the X3000 are not required, the X2000 is the best cost effective purchase. Review posts #3-6 on page 1 of this thread for helpful information to include firmware change log notes.