Have you noticed that all the cool kids on twitter have some sort of icon next to their name? And you want that also.
Here is the list of all emojis you can use in your Twitter profile name. It will show up as a cool icon.
All you do is select what you want on this list then cut and paste it as a text into your twitter profile name.
As you noticed they have banned all things that looks like a check mark because that is reserved for the verified people. Only really really cool people on Twitter can have that so though luck.
Finally it seems almost all emojis can be used in the twitter name and will show up as an icon as of late 2016.