How active is my mobile phone when I'm not using it?

Since iOS 10, I’ve seen very good ability to control activity. Even better, you can see exactly what runs, hour per hour what’s front most and what background apps are taking energy.

If you want no background - just turn that off for all apps. Then review the activity graphs for a couple days. If you want more activity, enable background and remove apps that are more active than you care or selectively disable background activity so you know apps only run when you ask them to run.

If you’re technically minded, the Charles proxy is excellent to log outbound traffic without needing to manage your router. I also use Ubiquiti network routers so know and log access / block some traffic, but that’s on the network side, not the iOS side.

I don’t think getting outside analyses are as important as choosing your apps so you know how they are funding development. If the app is slick and free, you’re paying them in data / marketing / eyeballs in many cases.

I wonder whether there is any study/website/research work or a place to check how much data my phone is sending when I'm not using it.

Yes. I found this research article (behind a paywall) that discusses these points from a battery and energy optimization point of view: Smartphone Background Activities in the Wild: Origin, Energy Drain, and Optimization

From the Abstract

• First, we present a large-scale measurement study that performs an in-depth analysis of the activities of various apps running in background on thousands of phones in the wild.

• Second, we quantify the amount of battery drain by all such background activities and possible energy saving.

• Third, we develop a metric to measure the usefulness of background activities that is personalized to each user.

• Finally, we present a system called HUSH (screen-off optimizer) that monitors the metric online and automatically identifies and suppresses background activities during screen-off periods that are not useful to the user experience.

https://dl.acm.org/doi/abs/10.1145/2789168.2790107