Help with DIY dedicated circuit

I also recommend an electrician.  One of the comments indicated to make sure both of the "HOT" wires to your two new outlets are on the same phase.   Your stove ran on "220/240 Volt" which means that, in the United States, the two "HOT" wires are just 120V that are out of phase.  When you wire these to the stove, the stove uses both 120V in a push/pull configuration.   You really should have an electrician do the work because you want either one single 20A breaker with both "HOT" wires leading out of that.  Or you could install two 20A single that are BOTH ON THE SAME PHASE to connect to your two "HOT" wires  of your romes 10/3.  However, there are definitely electrical codes to follow and I'm not sure of this particular configuration.  So, again, I highly recommend an electrician do the work and discuss the options with you.

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. 

After all this and watching the YouTube link, I am more confused. If I do the work, I think I will use the 10/3 wire that is in place already, but change the 30 amp twin breaker to a 20 amp single breaker only using one of the hot cables and capping the other on both ends. I'll piggytail either a 10/2 (will it fit in the wall outlet?) or a 12/2 cable from the junction box to only one 20 amp outlet and forget about installing 2 outlets at this time.....or I'll get an electrician....

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.