Is the velocity taken as zero at $t=0$ always?

I'm guessing you are referring to a problem involving conservation of momentum. Most of the times the required data are given. If the problem states that you simply drop the block on to the disc, then you have to take it's velocity as zero because it doesn't have any angular velocity at first. But just as when that block comes into contact with the disc, it will gain some momentum at that instant.

Of course in reality this is not the case. The block will be accelerated because of friction and it'll take some time to speed it up.

You take the conservation of momentum of the system before and after some change had occurred. In your case, before and after you place the block on the disk. Don't bother about what happens at t=0. Say you have a disk spinning about it's axis, and after some time, you drop a block onto it with zero initial velocity. Then the conservation of angular momentum states that,

$I_1omega_1+I_2(0) = I_1omega_2+I_2omega_2$

Where,

• $I_1$ = Moment of Inertia of the disk
• $omega_1$ = Angular velocity of the disk
• $I_2$ = Moment of Inertia of the block
• $omega_2$ = Angular velocity of the whole system after dropping the block

Hope that clears it up.