The word is:
The answers will also reference the title, in the sense that
A 6 is involved in each part
I'm a word with a certain number of letters:
Perhaps the first day for wearing new sweaters?
'Boxing' has 6 letters in it, and 'Boxing day', the 26th, could be the first day you can wear the sweaters you got for Christmas, the day before!
Putting in something with that number of sides.
As for what that number is, it's implied.
If you are 'boxing' something, you are putting it in a box, which has 6 sides
Another meaning, quite risky, some people shed tears;
Numbered Super World Classic for a couple of years.
Boxing is a dangerous sport, and the Super 6 World Boxing Classic was an event running from 2009 to 2011
To check out your answer, if still you're unsure,
It's also a county, but this one's obscure.
Boxing County is a county in China!
Happy BOXING day!!