Can London black cabs be booked to come and collect you like mini cabs?

Yes they can be booked in advance, hailed on the street, or picked up from designated taxi ranks.

Yes. There are dial-a-cab services for black cabs if money is no object.

Yes. You call the cab company and tell them when and where you want to be picked up. I don't know if it can be done online. That would be something you could check with the cab company.

Yes some have phone numbers on the side.
Only black cabs can be legally hailed in the street.

Can London black cabs be booked to come and collect you like mini cabs?