A list of all quests (including timed quests) can be found here.
A quest is a grouping of one or more objectives that, upon completion, reward the player with and either wages or . Where achievements are available to all players starting at level 1 and only have a single objective per achievement level; a quest can be either a single objective or multiple objectives. A quest can also require the player to be at a certain level, or have completed another quest, before the quest will begin.
There are no rewards for completing a quest. This is a frequent misunderstanding and covered in the FAQ.
When any objective in the quest is completed, the game will display success indicators. If the quest has only one objective, or if all objectives are met, the success indicator will be displayed and then the reward window will appear giving the player the opportunity to share the reward with the player's neighbors.
Since 22 January 2013, timed quests have become a staple of Megapolis. Timed quests are different from normal quests, and have a timed duration - usually 192 hours - in which the quest can be completed. After the timer reaches zero, the quest rewards can no longer be accessed, although the buildings and areas may still be available for construction. There is an option to extend the time by 24 hours for 3 Megabucks, which becomes available when there is less than 24 hours left on the timer to complete the quest.
Prior to Mar 23, 2013 these events did not repeat and there was no intention to repeat them. However, HawkeyeKeith suggested on Mar 23, 2013 that Social Quantum should repeat events for players who either started after the event ended, or for those players who were unable to complete the event within the time allotted. On April 2, 2013 the official position changed to "probably will be in the future (but not soon)". On April 8, 2013, the official position changed to "they will be repeated again in the future."
On August 10, 2013 the In Search of Atlantis quest, which had debuted on February 19, 2013, was the first timed quest to be repeated.
Since March 6, 2019 players can participate at missed timed quests by starting construction of the related buildings. A timed quest can be repeated provided that all TQ related buildings weren't purchased yet. It can run simultaneously only one timed quest whether new or old. Available TQs are in the fifth column of the table below. Since March 6, 2019 were 21 old TQs available. March 28, 2019 SQ added 8 old TQs and on March 11 another 8. Even new TQs (since "Apocalypse Park") can be repeated by demand (under conditions above).
If you want to repeat a TQ but it's not generelly available yet (with ✖️ listed), you can sell your buildings — only suggested when they're low developed and when all are sellable — to gain the possiblity to participate in future. It's not guaranteed that a specific TQ will be repeatable, so acting at one's own risk. Btw, to sell already purchased buildings will not make the TQ (with ✔️ listed) repeatable.
Timed quest rewards are not available after the event ends but that doesn't applies to all timed quests. However, contracts related to completing buildings may continue to be unlocked, buildings completed post-quest (where applicable) that generate coins will start to do so, and additional construction sites that spawn upon completion of certain buildings also continue to appear.
Here is a list (ordered by Xp level) of all feature based timed quest which instantly start when the player reached a specific level:
The Furniture Production timed quest starts as soon as new players reach Level 8; this TQ and the related buildings are unavailable for player who already reached Level 8. It was initially introduced in November 2016[Q 1].
↑The game files shows the date as February 20, 2017
↑SQ forums, Player Help & Support, Post # 718 - "We're not planning to repeat these events so far but that's definitely a good suggestion so I am going to transfer that to the developers. Though obvioulsy that will happen not in the near future."