Learn English – How “would prefer” & “would rather” used in combination with “rather than” when refer to another person

BBC English says

Rather than means instead of and can be used in combination with would prefer to and would rather.

My mother would prefer us to email each other once a week, rather than spend half an hour on the phone every night.

My mother would rather we emailed each other once a week instead of spending half an hour on the phone every night.

Let see simpler example,

I would prefer him to stay at home rather than go out (1)=
I would prefer it if he stayed at home rather than go out (2)

I am not sure the (2) is ok but I am sure (2) can be said as "I would prefer it if he stayed at home instead of going out" (3)

Thing is trickier when saying them in the past

I would prefer him to have stayed at home yesterday rather than (have) gone out (4)=
I would prefer it if he had stayed at home yesterday rather than (have) gone out (5)

I think (4) is correct but not sure if (5) is correct. But I think (5) can be said as "I would prefer it if he had stayed at home yesterday instead of having gone out" (6)

I would rather they stayed at home than go out (7) (not sure 7 is correct)

but "I would rather they stayed at home instead of going out" seems correct

I would rather they had stayed at home yesterday than have gone out (8) (not sure 8 is correct)

How "would prefer" & "would rather" used in combination with "rather than" when refer to another person?

Important Note:

Cambridge Dictionary says:

When we use rather than with a verb, we use the base form or (less
commonly) the -ing form of a verb:

Rather than pay the taxi fare, he walked home. (or Rather than paying
the taxi fare, he walked home.)

So, I think it is wrong to say "He walked home rather than paid the taxi fare"

However, another source said this sentence is correct "Ed walked rather than ran."