One question at a time...
I've done the calculation of 45KVA / 480V * sqrt(3) to roughly get 54.13Amps. Would this amperage value be how much the transformer can supply per Line ?
You have calculated the value correctly and this is the rated current value. This means that the transformer can deliver this amount of current without overheating.
Would L1, L2, & L3 all be capable of supplying this much?
Yes, each of the phases can supply that amount of current.
Then my second question is, If I add a 3 phase 10 HP and 3 phase 5 HP motor to the busbar connecting to this transformer. How would the amount of amperage the transformer can supply change?
If the motors are connected to the 600V bar, the same one that the transformer is connected to, as you comment. In this case, the transformer will be able to continue supplying the same amount of current, the rated current in principle, without interference from the motors. However, if your 'source' is not powerful enough, you may have problems with voltage drops and, perhaps, with the motors starting in the case of direct starting (this will depend on the impedance of the source, that is, the equipment that feeds the 600V bar).
If the motors are connected to the 480 V bus they will certainly draw the current from the transformer. As you mentioned, about 15.54 A, which corresponds to 28.7% of the transformer's capacity. Nothing prevents you from drawing more than the rated current, but in this case the transformer will probably overheat. Now, special attention must be given to starting the motors, if you are using direct starting, because the series impedance of the transformer will cause additional pressure drop.