How do battery packs split power?
Power supplies don't "split" anything, they just maintain their output voltage, and can supply current up to their specified maximum power capability, whatever it is. The loads are connected in parallel, and each load will take whatever current it needs. 

If you have a 5-V supply with 2.4 A rating, the fan will take 0.35 A, not more. Fan's rating of 5V@0.35 A means that the fan has resistance of about 14 Ohms. The 14-Ohm resistor will take 0.35A from 5-V supply, according to Ohm's Law.

If there is another load, say 5 V 1 A, then it will take its own 1A, so the PSU will output 1.35 A total.

If the PSU says 5V 2.4 A, then it can't deliver more than 2.4A. If you try to connect a load  rated as 5V@6A, it means that the load has 0.833 Ohm resistance. If you will try to connect 0.833 Ohm to 5-V output, it will try to take 6 A. The PSU will just collapse and output nothing in best case, or smoke out in worst case. 

In sum, the output current of a PSU will split itself between multiple parallel loads in accord with their corresponding load resistances, but only up to the PSU design capability.