Attaching small, nearly zero weight cable clips directly to drywall without anchors
Keep in mind what you're dealing with here.  Drywall is chalk between 2 pieces of paper.  Here, take a piece of chalkboard chalk. Soak strips of newspaper in 50/50 elmers glue and water mix, and roll them around the chalk.  Let dry.  Congrats, you've made drywall. 

However, fragile as it is, an ethernet cable is a trivial load.  It'll be fine. 



Just so you know, the National Electrical Code does have a chapter on non-PoE Ethernet wiring, it's in chapter 8 (800 et.seq.) And better, as Section 90.3 says, “Chapter 8 is not subject to the requirements of Chapter 1 through 7 except where the requirements are specifically referenced in Chapter 8.” So there are several ordinary requirements you can ignore.  

However, chapter 1-7 requirements which prohibit data cables still apply.  You cannot use mains wiring or conduit as a hanger to support data cabling.  You cannot run data cabling inside a box or conduit containing mains wiring.  Code requires mains wiring to be able to contain an arc-fire event where insulation melts; and if a mains wire melted into comms wires, obviously it would put mains voltage on equipment not rated for it, which would spread the hazard.

If you're doing PoE, depending on the load that may push you into Class 2 or Class 3 wiring methods, but that's beyond the scope of your question.