As you already found out, there is no direct way of finding download numbers for a specific app.
Some companies disclose such information when milestones are reached or within interviews and blog posts. Did you try searching?.
If you're just interested in how competitive a category is, for ASO purposes, you can checkout app worth per category in SensorTower. I doubt the actual figures are accurate but it will at least give you an idea of how they stack against each other.
Alternatively, you can look for 3rd party articles like this: http://techcrunch.com/2013/06/27/how-do-you-break-into-iphone-app-store-top-50-try-23k-free-daily-downloads-950-paid-or-12k-in-daily-revenue/
Let me know if you want more help with you specific case.
Here are a few services that are going to be your best bet:
I'm not sure how accurate these are, but I think these tools will enable you to do the research you need to do.
No. However there are a number of services that analyze Appstore data and attempt to guess the number of downloads for specific apps and categories. Most of the detailed information is pay-only, keep in mind for iOS apps, this is all based on guesswork since Apple does not release detailed data for the entire store.
For the education (free) category, an app that I worked on reached the top 25 with ~1,000 downloads a day. The bar is higher now across the board.