I'm not sure if this should be a guide or a forum post—i'll move this if needed. I couldn't find this information anywhere in the current Steam forums and (English) guides, so i compiled my own. Man, that was painful.

There are the gold badges, and then there are these. The game doesn't really tell you about them or what their requirements are for each level, so all of this data is considered spoilers. You've been warned.

These badges are necessary to get 100% of the achievements. They're also useful for gauging your pace for the Beat the Devs achievement, although the achievement is possible with just gold times. See this guide for more on that (and compare with the overall total here).

Note that, since i got this data experimentally, i could not get the time limit for the last level, since it doesn't give the usual results screen when completed. The time you see there is just what i had when i compiled the data. (I'll update it if someone can provide it.)

• Deaths: the number of times you die in the level must be less than or equal to this number.
  
• Time: the amount of time you take to complete the level must be strictly less than this. All times listed consist of a whole number of seconds, i.e. xx:xx.0, unless specified otherwise (i.e. the last level).

Note that the timer in the bottom left (if the on-screen timer setting is checked), which as far as i can tell matches that on the overworld if you just set a new best, is about 0.05 seconds higher than the time shown on the right in the results screen. The latter is what matters for the badges.

Note also that all of this is true for the gold badges as well.

World 1 total time (without Bonus):

World 2 total time (without Bonus):

World 3 total time (without Bonus):

World 4 total time (without Bonus):

Overall total time (without Bonus levels):
43:52 + ??:?? (between 45:45.9 and 47:12.0)

If you don't like hovering over every spoiler tag, you can instead view them here.[docs.google.com]