Can a DPDA be created for a Language which includes the null string? Can a DPDA be created for a Language which includes the null string?

The way I understand it, in a DPDA, you can have at most 1 move from any instantaneous description Delta(q,a,Z) where q is a state, a is some input, and Z is a stack-character.

Also, if you have a transition Delta(q,lambda,Z), you can't accept any other input from that state.

So, back to the original question. Can a DPDA accept a language which lambda and anything else?
Yes, we can simply make the starting state an accept state. With no input (i.e. empty string) the machine halts immediately, and accepts.