Hi,
Rich people used to, and still occasionally do, invite their friends out for an afternoon of shooting pheasants.'To wing a pheasant ' means to wound it. I guess you'd do this by hitting it in the wing, since if you shot a little bird in the body, it'd be dead. 'To wing someone' has come to mean 'to wound, usually in an arm or leg'. Western move dialogue:
Did Jim get killed?
Nope, the varmint just winged him.
'To give an at-home' means to tell your friends that you will be at home on Sunday afternoon between 1 and 5pm, thus inviting them to come and visit you.
In terms of social activities, both of these expressions seem old-fashioned to me.
Best wishes, Clive