Is coil current the same as load current?
It's the load current for the transistor (8mA, see below). The load current for the micro is the base current (0.8mA, see below). The ultimate load has current that goes through the relay contacts.
Relay 45-191 from Hosfelt.com has a coil current of 12/1500Ω = .008 A and switches 0.5A so it has a current gain of 0.5/.008 = 700. The relay load current is 0.5A.
What does base current mean? The current that must go into the base in order to get collector current to pull in the relay. Relays and transistors are current operated.
The pins are analog output pins at 8.8mA and 5V.
So to get a collector current of 8 mA and
assuming the 5v does not drop very much when it sources 8mA and
assuming a 1v base to emitter drop,
you need a base current of 0.8 mA and,
so (5-1)v/.0008A = 5kΩ for the resistor shown in your diagram.
Can each part in the circuit have many different rating possibilities – depending on the other parts?
Yes, they are mostly interdependent, so start at the load current and determine the rest of the parts from there. However, generic relay driver circuits may cover 90% of the applications. The ULN series ICs interface from microchips to loads like this.