As I understand it.
Originally RJ connectors were designed for telephone standards. One design feature was you could put a smaller plug in a larger socket and it would connect to the first pairs. To achieve this the first pair was on the center pins, the second pair on the pins straddling the center, the third pair straddling those and so-on.
Unfortunately this was not good for high speed data. The splitting and straddling of pairs was bad for signal integrity and this got worse as the pair count grew. However there was a desire for backwards compatibility with basic phone applications. This resulted in a compromise pair layout. The first pair was in the center, the second pair straddled the center, but the third and fourth pairs did not straddle any other pairs instead being placed with one pair down each side.
10BASE-T implementations chose to use this layout and chose to use the Second and Third pairs. Presumably the second and third pairs were chosen in an attempt to allow coexistence with basic voice service on the first pair.
Ok, that explains the pin allocations but what about crossovers? Well 10Base-T and 100Base-TX use one pair in each direction. For correct operation the transmitter at one end must be connected to the receiver of the other. That means we need to cross-over transmit and receive somewhere. Most of the time this was done inside the equipment, network cards used "MDI" pinout while hubs used MDIx pinout. So they could be connected with a straight cable. From time to time though it was nessacery to connect a MDI port to a MDI port or a MDIx port to a MDIx port, so a crossover cable was needed.
Later auto-MDIx came along allowing automatic switching, so it did not matter if you used a crossover or straight-through cable.