Apple products interconnectedness (Apple TV and Siri)

a) After enabling AirPlay I can control my Apple TV box with my iPhone or Apple watch. Correct ?

AirPlay is a separate function, and not a prerequisite for the software remote function. Both are possible independent of each other. Set up the Apple TV Remote on your iPhone or iPad - Apple Support

b) iPhone to TV screen-sharing is enabled via Apple TV box correct? (I suffer with broken miracast on my TV at the moment)

Yes. Use AirPlay to stream video or mirror the screen of your iPhone, iPad, or iPod touch - Apple Support

However, AirPlay supports also depends on support by the content provider. Some may choose not to support it. That is not a real issue if they have a native app for the Apple TV box.

c) Apple TV Siri is not supported in my region (CZ) so I will not be able to manage any apps or settings via Siri, BUT it can still be used to enter queries into search. Is my understanding correct? (I speak in english most of the times so I would configure it to CZ region but language to English if possible)

Dictation for text input fields is supported in many more countries than Siri digital assistant questions and commands. Use Siri dictation on Apple TV - Apple Support

d) I use Siri quite frequently for timers, reminders, notes, music and calls. My ideal vision and reason why I would like to buy Apple TV set is, when at home and connected to home wifi, to be able to ask Siri on my iPhone to play music from Soundcloud/Spotify from speakers connected to TV (Apple TV box ?) or to play a film from Netflix/HBO on my TV. This is the the core of my uncertainty at the moment - will I be able to ask Siri on my phone to handle my TV or apps on Apple TV ? (to switch on the TV I guess not so lets consider TV is always switch on for the sake of simplicity)

Siri on iPhone works for your iPhone. Siri on Apple TV works on Apple TV (in select countries). Those are quite separate. For Siri on Apple TV, use the Apple TV Remote. Use Siri on your Apple TV - Apple Support.

Siri on iPhone won’t help you circumvent the lack of Siri for Apple TV support in your country, which is mostly about which services are available there.

Netflix doesn’t allow AirPlay anymore, so you can’t use Siri on iPhone to stream Netflix content to the Apple TV box. Other services may support AirPlay commands from Siri.

Personally, I prefer clicking through a GUI over speaking to a device.

Why not find out for yourself? That’s what the cash back return window is for.