Help with DIY dedicated circuit

You can use 10/3 with a 20 amp you can always go larger gauge but never go smaller. 

You also need to make sure the new 20amp  double pole is on different phases in the panel box. It should be if the old 30amp was installed correct. 

You're not suppose to use the same neutral wire for 2 seperate breakers you can for a double pole since any overload on the neutral will trip both. 

Read your own link jea48 it clearly states to use 2 PHASES ,  I suggest you quit giving advice you are past clueless. If he puts 2 20amp breakers on the same phase using the same neutral wire then the neutral could carry 40amps  his 10/3 existing wire is rated for a 30 amp load which could cause the neutral wire to overheat causing a fire. The breakers need to be on seperate phases where they balance and the neutral carries the difference between the loads, on the same phase the neutral would carry the sum of both.  You can use 2 seperate 20 or 15 amp breakers running one hot black and one hot red  and both use the same white neutral as long as those breakers are NOT on the same phase. Code is to use a linked 2 pole breaker IF it's a 240v  single phase OR multi- wire branch circuit with a SHARED  neutral. The OP needs to call an electrician mental is right  to much nonsense to confuse him. 

Pigtail with 12/2 it's easier to use and is fine for a 20 amp receptacle. Most likely the old 30 amp breaker is on 2 phases and you would be fine running 2 new breakers but unless you know for sure do what you said. 

Sorry he deleted his posts it looked as though you were directing it to me as I didn't see his post. I can see I made a mistake going back and reading it again.