Enclosing cable run through living space
You can't use cable management systems. They are not listed for protecting mains wiring.  At best they're listed for extension cords, but probably not.  

You can use UL-listed surface conduit such as Legrand Wiremold.   You can also use regular conduit if you don't mind the industrial look and chicken-choking the cable down the conduit. I recommend EMT because it's not bad to work with, and is easy to correct or modify. 

Either way, you can use the conduit in one of two modes:  


Using the NM cable wiring method, where you run it in the normal way you run Romex, and enter conduit ad-hoc (no junction box) merely to provide a damage shield where the cable may be subject to damage.  In this case any conduit that fits over the cable will suffice. 
Using the conduit wiring method with continuous conduit between junction boxes. Here, if you insist on using cable, you must use 3/4" conduit*, due to conduit fill rules.  The pull will also be quite difficult.  However you are better off splicing to individual stranded THWN-2 wires, in which case you can easily get away with 1/2" conduit, and have an easy pull too.


For a 20A receptacle and breaker, you are perfectly allowed to use 10 AWG wire.  However, you don't need to... you can do it with #12 wire.  At 230V, #10 is best if you are going over 150 feet. 

Also, for an air conditioner (NEMA 6-20 outlet), you don't need a neutral wire.  However, including a neutral wire will allow you to convert it to two 120V circuits later.  If you wind up doing the entire run in actual conduit (EMT or Wiremold), you can run individual wires, omit the neutral wire for now, and add it in the future when you need it.

If you use metal non-flexible conduit, that is a valid grounding path (indoors), which means you don't need to run a ground wire.  My air conditioning circuit is simply 1/2" EMT conduit with a black and red THWN-2 wire inside. That is it. 

* I'm assuming you are using round 10/3 cable.  If you're using flat cable, measure its widest dimension. multiply by 1.37, and you need to use conduit whose inside dimension is larger than that.