Hello Viv, yes, I think you are quite right when,

you said,
I wonder if there were people who, in effect, fostered children [possibly
illegitimate or orphaned children??] If not what would have happened to those
children?
Did either Catherine Quayle or her mother Margaret Jane Quayle [nee corris]
look after such children?

on looking again at the 1901 Census, Charlotte Murphy Boarder S 34 Isle of Man Michael - is noted as being an Imbecile.
I agree that their Boarding House, was used to look after illegitimate or orphaned children.

you said,
William Quirk Quayle was born 1879 while William Quayle Cowin wasn't born
until 1887 and his age is consistent with this in the 1901 and 1911 census so
I'm not convinced this was the same person.

I am not meaning that they are the same person, I was just showing William Quirk Quayle as being the son of Thomas Quayle & Margaret Jane Corris.

you said,
I suppose it is a possibility for Margaret Corris [mother of William Cowin] to
be a realtive of Margaret Jane Corris.

Yes, I agree this could be a relative of Margaret Jane Corris - would their be any presentments for William Quayle Cowin born in 1887 and is the Parish Register worth checking to see if there is any added information ?

glad to have been of help,

Averil