1) A "Repeater channel" is a combination of two channels - a sending channel, in the 467Mhz range and a receiving channel in the 462Mhz range. The receiving channels are shared with some of the 'regular' channels, so you can hear someone talking on Repeater 24 on CH16, but if you transmit back on 16, they (likely) wont hear you. When you put both a TX and RX tone on Ch24, you are setting the radio to only listen to Ch24 repeater traffic. If you want to listen to ch16, put the radio on Ch16 OR, dont enter a RX tone on Ch24
2) Depends on what "emergency channel repeater 20" you are referring to - But in general, you must enter the correct TX tone to use the repeater. As eluded to in #1, adding the RX blocks out all other traffic except for traffic using that tone - so if the repeater is transmitting with a tone, if you DONT enter that tone you would still hear the repeater, and you would also hear anyone else talking on that shared channel. If the repeater does not transmit a TX tone, and you enter a tone (or enter the wrong tone) you wont hear the repeater - you will only hear transmissions that are sending the tone you've entered.
Since you're new to GMRS, it is my job to make sure that you also know about Channel 19 (no tone) which many people use as the "unofficial official" road and travel channel, and Channel 16 (no tone), which many people use as the "official" off-roading/hiking/adventuring channel. Also be aware that "some people" will prefer to argue over the use of those channels rather than actually do/say anything helpful or constructive - just remember that nobody cares what the complainers have to say so you can ignore them - dont worry, they'll be easy to spot.
Welcome to the forum and welcome to the exciting world of GMRS!