I want to check iPhone's battery charge level from my MacBook Pro. Is it possible to do so?
There's a neat trick that lets you check the battery charge level/percentage on your iPhone even without touching it.
For this to work, you'll need to:
Make sure you have Show Wi-Fi status in menu bar setting enabled under System Preferences app → Network → (active Wi-Fi network).
Now you have the Wi-Fi icon in your Menu bar status area. Press and hold the Option key on your MacBook's keyboard and click on the Wi-Fi icon in the Menu bar. In the context menu that appears, your iPhone will show up under Personal Hotspot. If the Personal Hotspot or your iPhone doesn't show up, just make sure both Wi-Fi and Mobile Data is enabled on your iPhone.
The entry corresponding to your iPhone will show the battery charge status, similar to how it's shown in your iPhone's status bar. The battery level indicator icon is shown in red if the charge level has reached below 20%.
On hovering the cursor over the iPhone name, you can see the exact percentage of battery charge level in a popover view.
Do note that this trick works only when you press and hold the Option key while clicking on the Wi-Fi icon.
Is there any macOS app that acts as a client for my iPhone on my MacBook Pro?
I'm not sure what exactly you are looking for here. As far as I understand, there's no dedicated iPhone client app in macOS.
Not only would I like to check iPhone's battery level from my MacBook Pro, but also launch or close other apps, change settings, etc, on the iPhone. Is it possible?
Generally speaking, it is not possible to control your iPhone from your MacBook. It's not possible to run/close installed apps and send multi-touch gestures to iPhone remotely.