why my profile has this odd icon?

Question

Highlighted in the image below. This icon appears in all sites with my account:

Answer 1

If you click on that icon—it's actually a hat—you'll encounter a pop-up that then provides a link to the Winter Bash site.

Every year, StackExchange runs this Winter Bash, where users can collect hats for completing tasks on the site. You can see some of the hats on the link above, along with the descriptions of how to receive them. There are also many, many "secret" hats that aren't shown there.

As of this year, you can choose to opt out of the Winter Bash by clicking on the snowflake at the top right of the browser window and choosing "I hate hats."