Let me first say that my background is theoretical physics so I find it hard to look at some of the mathematical literature.
The kind of problem I am interested in is the following one. Consider a 4-dimensional smooth, simply connected orientable manifold M and suppose A and B are immersed surfaces. Denote with $A\#B$ the connected sum of A and B (since M is simply connected I think there is only one way of doing this). Denote with $A\cdot B $ the intersection number of A and B in M. Is it true that the self-intersection number of $A\#B $ in M is given by $(A \# B) \cdot (A \# B) = A \cdot A + 2 A \cdot B + B \cdot B $?
If A, B are 2-spheres, looking at Freedman and Quinn, Topology of 4-Manifolds page 24 I would think so, but I am not sure I am understanding the book well. What could be said, perhaps making additional assumptions, if A and B are surfaces of more complicated topology?