Main Story: You are forced to do this in the tutorial
Remove cumulative; Add Level
Shop -- you have to buy the activities with gems.
Level -- you must level up to be allowed to own more pets
Time/Date Sensitive: Those are 'real' days of the week
Shop, Level, Time Sensitive -- earning all of the awards requires unlocking the activities in free play and a handful of awards are have a minimum level of 5 or 10 to be able to win. The awards for talent shows have to be done on specific days of the week as well.
(I think you can probably get 75% w/o levelling up.)
Time/Date Sensitive: Main Events are only on Saturday (real time)
Shop -- you have to unlock them with gems.
Shop -- it costs 50 gems each time to change features (or more gems if you're unlocking a new feature through the atlas)
Time/Date Sensitive: Strike a Pose events happen only during a talent show. Each one has 5 events and you can only do one talent show per day so this requires doing 10 events in a row spread out over 2 (or 3) consecutive days.
Time/Date Sensitive
Time/Date Sensitive
Time/Date Sensitive
Cumulative: Doesn't have to be all at once (and each 'chase' event lasts only about 20 seconds).
Remove cumulative; Add Level
Remove cumulative; Add Level
Remove cumulative; Add Level
Remove cumulative; Add Level
Remove cumulative; Add Level
Remove cumulative; Add Level
Remove cumulative; Add Level
Remove cumulative; Add Level
Shop (the only way to get rid of gems is to spend them).