私 General - and gender neutral
僕 Young boys use this (very young... like when they start to speak up through an indeterminate age).
俺 Young (adolescent) use this as well, but it's a more haughty.
You'd never hear someone use this in a professional setting.
You'll almost never hear young boys using 私 among peers.
私 As above, but common for girls to use it among peers.
あたし Female only, relatively more flexible than the male counterparts.
うち Comes from 家 (meaning home or family).
So yeah, you're literally saying "My home's cat" or "My home's mother".
You could go on for days as to why this is but it's likely due to the
weight put on the family unit and not the individual.
わしゅ sounds like slang for わし which you'll probably not a hear
a lot outside of TV but perhaps you might. The take home lesson is,
as you stated, it's for the elderly. :)