1)"Do you mind his calling on us?"
OR
2)"Do you mind his coming by our place?"
OR
3)"I don't object to his calling on us?"
OR
4)"I wouldn't say no to him/his coming by?"
Are 2),3) and 4) correct?
Do you mind him calling on us?
Do you mind him coming by our place?
I don't object to him calling on us.
I wouldn't say no to him coming by.
All these can be used. Be careful with him/his; his is the possessive pronoun.
Again, him.
His book, his car, his phone; you would not use him in those circumstances and take note of my previous comment.
Actually both are possible!
"You don't mind my calling you Tony, do you?"
"We discussed Alan's leaving all of his momey to charity."
"I won't risk their being arrested."
However, many speakers of American English regularly use nonpossessive forms of nouns and object pronouns instead of possessive forms. Thus you will also here
"You don't mind me calling you Tony, do you?"
"We discussed Alan leaving all of his momey to charity."
"I won't risk them being arrested."
p. 504 Ron Cowan "The teacher's grammar of English" 2008