Aharon , you give me a complex ,,,,
As i do not keep my guague samplers , i use them in my art or give
them to people who can use them , or incorporate them in some crat
thing ,,,,
Knitting needles are sorted to Dp , circular and plain pairs , each is
in a plastic Office drawer [ part of a pile or Totem of office
drawers..
Left over threads ,, if enough are going into plastic bags with same
group of wool , i am sorting by group by colors .
Small amounts i store in plastic jars ,,that are shelved quite high
,,makes it easier for small odd pieces that i make ,,,,,
As to patterns ,,,, My own patterns are written down in copy books
according to date i fill the copybook page after page ,,,
Ready made patterns are in their mags or books ... i have a card index
that goes by Subject and if i used something or read some pattern i
will make a note ,,,,, xxxmag , date , page ...

When do you just throw the leftover yarn balls to the cats to play with?
never ,, but i am sending some leftovers to colleagues who can use
them ...
Do you sort by yarn thickness or by color?
Thickness and type ,,,,,
I keep projects in progress in clear plastic shoe boxes for small projects
I sewed somew big Cloth bags [ sheeting materials leftovers as well ]
and hung it on coat hooks on the wall , and have a cloth wash basket
in which i have small bags with [ still] small works ,, The loom keeps
it`s on work on !!!

What do you do with your bits of left over yarn? Is this where the squares
exchange comes in?
Small cellophone bags , adorning bigger works , into various art
projects , Wool balls , hair adornments , dolls , A vertical striped
sweater or vest ,,,,,
You might consider making a Left over blanket or cushion ,, just work
what ever left over pieces you have into that blanket ,,, after a
while without any effort you will have a beautiful SINGULAR blanket

Should RCTY work together to get into the Guiness Book of Records by
createing the world's largest ball of yarn, one left over bit at time