According to various Googled sources, it IS allowed to use 15 amp receptacles on a multi receptacle 20 amp line. There are conflicting statements in the NEC code. But the consensus is the multi 15 duplex on 20amp branch is OK per NEC. On a one only single duplex for a 20 amp branch, no. that outlet should be a 20.As for using a 20amp for a CDP. Who cares? A CDP is using 20 to 40 watts, maximum. The power cord for the CDP can be 20 gauge magnet wire, like Mapleshade.. Or a 7 gauge Pangea.
You can have a 20 amp circuit with one 5 watt light bulb. The bulb uses 5 watts, whether the cord it uses is capable of delivering 20 amps or is only some 22 gauge magnet wire. The light bulb does not care.I would add, if you are so uncertain, do NOT do it. Find someone who can deal with it. Making mistakes when it can be dangerous? Just do not do it. Or at least find out enough to feel confident.Fiddling with stuff you do not understand is not a good idea.