How can I know the top dowloaded apps in my country
How can I know the top downloaded apps (by category, say games, messaging, etc.) from the App store in my country?

Open the App Store app on your iPhone, iPad or iPod touch, navigate to the Apps tab and scroll down to access the section where you can see the overall Top Free Apps and Top Paid Apps as per the recent download statistics across all the categories.

The data is shown for the country associated with your iTunes and App Store account and it differs between the App Stores from different countries.
You can further scroll down to access the Top Categories section and can tap on the See All link to see all the major app categories.

You can then tap to navigate into a desired category to view the Top Free Apps and the Top Paid Apps in that category. The below example shows Top Free and Top Paid apps from the Education category.


How can I access Apple's Kid's App Store?

There is no separate Kids App Store, but Kids is available as one of the category within Top Categories. You can browse to the Kids category to see curated kid friendly apps.