Minimum total waiting time for arrivals/durations
Yes, this is NP-hard. Since neither the $a_i$ nor the $d_i$ depend on the solution, the problem is equivalent to the problem of minimizing the sum of the completion times of the requests, which is known in the scheduling literature as "$1|r_i|\sum C_i$" and is NP-hard. Reference: 

J.K. Lenstra, A.H.G. Rinnooy Kan, and P. Brucker.
Complexity of machine scheduling problems.
Ann. of Discrete Math., 1:343-362, 1977. 

Also, I disagree with the statement that "the only difference to Interval Scheduling is that here all clients have to be serviced". There is one more important difference: in Interval Scheduling, the algorithm can just accept or reject the clients, while in this problem, even after the algorithm accepts a client, it has to choose when to start it.