Which sidequests need to be started before the end of the main storyline?

Yahtzee is wrong; no sidequests need to be started before the end of the main storyline. The lead-ins to some sidequests disappear due to story events, such as Hot & Cold and Through the Looking Glass, but it's still possible to complete them.

Hot & Cold is started by talking to Harley during your second visit to the Steel Mill. She isn't hard to spot. She lets slip that there's something helpful in the basement of the Steel Mill. Through the Looking Glass starts from a radio broadcast from Alfred after saving Vicki Vale. There are reports this one may disappear, but will come back if you complete the other sidequests.

One achievement is missable if you complete the sidequest where it can be completed, namely Catch.

Some sidequests are on timers and take a while to advance, particularly Shot in the Dark. Identity Thief can be done at any time after the first corpse is found, if you explore on your own; the feeder quests seem to only appear if there are no other sidequests in the vicinity. The Enigma Conundrum is also best started when it becomes available, after Batman visits the Steel Mill the first time, as the feeder quest for that takes a long while to appear.

The Enigma Conundrum is the Riddler sidequest, and starts at the church once Riddler puts his banners up. After answering the Riddler's question, he will announce that he has informants all over Arkham City; this will allow you to interrogate them through the rest of the game, if you'd like hints on trophy locations. Like the sidequest itself, Riddler informants aren't missable and will respawn if you knock them out instead of interrogating them.