I am running cat 5e do you need a conduit chase going throug a single wall?

Answer 1

Not in my area's building codes, at least in residential installations.

In some commercial situations it might be required - I had one where a false ceiling was actually an air plenum and part of the HVAC system. According to the code, ALL wiring that runs in a plenum must be in conduit, even low voltage network and telephone wires.

Answer 2

No. When we rebuilt our house last year, we ran an entire data network using cat5 to every room in the house, terminating at a network hub in the back of a closet. All of the network outlets (two in every room of the house) are activated through a router and we don't have any problems with the network. It passed inspection, too.