Design

Are you talking about the water supply lines located between the water storage tank(s) and the sinks & showers? If so, you don't need a trap on those supply lines because they'd be normally under pressure.

There should be no physical connection between the water supply lines (hot and cold water) and the drain lines! If there is a physical connection it is a code violation and a health risk. There is normally a vertically-oriented air gap between the two types of lines.

In regard to the drain lines, yes, there should always be a tap and vent stack located in the drain line to remove the sewer gases.

Check with your local building code enforcement officer/inspector and thoroughly familiarize yourself with the state building code and it's requirements and plumbing standards.

Uggghhh! I can't believe I answered this OP!