Do infants have increased immunity right after infection ie they may not get infection from same virus if not recovered frn another one in recent past?

Yes, but specific: Any infection stimulates immunity to that specific virus, bacteria, etc, and makes it harder to get the same infection twice, however organisms adapt and mutate, so you must still be careful not to expose to the same infectious agent. Overall the immunity to that cirus or bacteria is increased

Yes: Anyone that has a normal immune system, including infants develop selective immunity to a variety of germs than pass through town.Viruses generally trigger good permanent resistance to the same strain, while bacteria often do not.In other words, they will not get the same stomach virus again from the family member they pass it to, but they might pass strep back and forth.