After this latest update I thought I'd run the numbers to see how many feathers are possible based on what strategy is being used. Thought you guys might be interested, so I've summarised it all below. Let me know if you see an error, have a question, or want me to explain a step in a bit more detail.
Spreadsheet
To see the details of all the below calculations you can view or download a copy of my spreadsheet
here.
Total SE
With these new quests available we are now able to get a total of around 6671 to 6991 SE over the course of the event, assuming you complete all quests, and taking into account the variable amount of SE you can pick up around your city. I've lifted these numbers straight off ElvenStats.
To figure out how many feathers we'll get we take the total SE we'll get, and divide it by the average feather cost of all the chests we select. To calculate this we use the effective 'Sky Essence Per Feather' figures found in the 4th column of the table at the very bottom of
this page and then adjust it based on the probability we'll actually pick that chest. The latter will depend on what strategy we use.
Strategy 1: choosing a random chest every time.
If we pick our chests at random each time then we can assume every chest will be picked roughly the same amount of times each, so the typical amount of SE spent on each feather works out to be the average of the effective feather costs of the 9 available chests, which is 23.24 SE per feather. This is the same figure ElvenStats uses in their own calculations to figure out how many phoenix artefacts will be obtained. This results in 287 to 300 feathers, or 7 to 8 artefacts. This method will also net you about 19 or 20 daily prizes.
Strategy 2: always choosing the most cost effective chest for feathers.
This method will maximise the amount of feathers you get. Essentially it involves always picking the chest with the cheapest feathers (see above link to table on ElvenStats). Since some chests will be picked much more often than others this calculation is non-trivial. It involves figuring out every possible combination of 3 chests you'll get, then figuring our which chest you'll pick with this strategy, then taking an average of the feather cost of the chosen chests for all those different scenarios. This works out to be 20.26 SE per feather, which will get you 329 to 345 feathers total, or 8 to 9 artefacts. Note that 9 is enough to upgrade a single phoenix to the maximum level. You can also expect to gain about 19 or 20 daily prizes with this method.
Side note on Strategy 2:
Previous calculations by myself and others have used a much simpler calculation, which involved simply taking an average of the 7 best chests (assuming the 2 worst ones will never be picked). This predicted 300 to 315 feathers, or 8 artefacts. I now believe this calculation to be incorrect, as it doesn't take into account the frequency with which some chests will be picked. Just something to be aware of in case you find these numbers floating around still.
Strategy 3: always choosing the most cost effective chest for daily prize.
This method will maximise the amount of daily prizes you get. Since not everyone is focused on the grand prizes I thought I'd run this scenario as well. Essentially this method involves picking chests based on the cheapest 'SE Per Daily Prize' value (column 5 in the table linked to above). This calculation is similar to Scenario 2, except a different chest is chosen from all the various possible scenarios. The average feather cost of these chests works out to be 23.41 SE per feather, which will get you 285 to 298 feathers, or 7 artefacts (maybe 8 if you're lucky). You can also expect around 25 to 27 daily prizes.
Summary
Assuming you have average luck, and you complete all quests, you should pick up at least 7 artefacts regardless of which method you're using, which is enough to get a level 8 phoenix. If you pick a strategy to maximise the amount of feathers you get then you can potentially get enough artefacts for a level 10 phoenix without needing to use diamonds. You can also expect around 19/20 daily prizes if you either pick chests at random or optimise them based on feathers, or as much as 25 to 27 if you're instead focusing on daily prizes.
I hope you guys find this useful.