To use an armor piece as costume you have to convert it into a costume piece first using a 'costume transformation elixir' (either buyable in cash/shell shop or as login reward).
The transformed piece will loose all stats but gain a percent value ranging from 1-10% (depending on what transformation elixir used - if you can use 10% for best result).
And yes, you can store and switch between 10 costume sets.
Furthermore the round slots next to each costume slot is for shards to be equiped.
The first slot is always free, the second unlocks once you have 10 costume pieces of one kind (eg. hats) equiped.
As for the value of shard that can be used depends on the aforementioned percent value on the costume piece as well as on the 'energy' value of you current armor (the light blue stat).