CTCSS is continuous tone-coded squelch system and DCS is digital code squelch. They can be single transmit or receive or both on TX and RX. The difference is CTCSS is analog (and is the same thing as PL Tone, which is Privacy Line, a Motorola trademark) and DCS is digital codes instead of a sub-audible tone.
Split tone just means you have a different tone or signal on TX than RX. That's not unusual on repeaters because a listener can scan for the tone being used on the output that you hear and if you use the same tone on input they can maybe sneak in unauthorized. By using a different tone on input you can prevent this. To get into the repeater someone would have to try all possible CTCSS/PL/DCS. Tedious at least.
Having a tone on the output of a repeater is only for courtesy, you will always hear the output tone or not. If you use a tone to open your squelch on RX it's usually because you have interference so it's double step to prevent static. You only really need to be able to add a tone or DCS to your TX signal if the repeater is set to require it.