Why both T568A and T568B standards

The thing you're looking for are called Baluns:

For Distributing Audio/Video Via Cat5 Cable

Today's A/V industry has created the need to send high resolution audio and video signals over increasingly long distances of cable. However, the size and cost of the video cabling, especially when long lengths are involved, had created a challenge. As needs and technology change, future cable replacement can be quite costly.

A solution has been developed to tackle this problem. By placing "balun" adaptors at each end, signals can be adapted to be effectively sent over long distances of Cat5 cable.

i found a company that has a whole product page of them.

One of the product pics has this schematic on the actual device:
_{(source: mcmelectronics.com)}

So you can steal their pinouts and use that.

I believe that the difference between 5 and 5e is a more specific description of the twists per pair and twists per foot of each pair.

Cat5a and Cat6 further enforce the geometry of the pairs relative to each-other over the course of the entire cable and restrict the untwisted amount at the ends.

In practice, if it says Cat5, it is an old cable. "Cat5" hasn't been made in years, and any twisted pair cable suitable for use in networking (ie not cat3 or something used for a thermostat or lawn sprinklers) will be a Cat5e or better.

A true Cat5 installation is sufficient for gigabit ethernet. I've seen plenty of things labeled cat5 that are not actually up to spec for one reason or another, that run 100mb fine but don't work with gig.