As far that I'm aware, it is key management/distribution problem and how to expose it to the user. So in order to not manage them, only the device with the phone number that the account is registered with can do it.
Desktop apps are not the ones where you create Telegram account; you only authorize them.
For another example, Threema, which is E2EE by default, doesn't allow more than one device with the account (to use multiple devices, you must create several accounts, one for each device and link them, and then basically any chat is a group chat. For desktop use, the web client uses webrtc to talk to your device to use it as a proxy).
1. Does not really matter to the argument; you cannot do E2EE on the desktop with such an account. But you could on the mobile device, once you install the app there. They call it "device of origin".
2. Which one? There are several and the one I'm running (the official one; brew cask telegram-desktop) does not.
3. They are third party and if it doesn't work nicely or has missing functionality ("why are not my chats from other devices there?"), Telegram devs can still point fingers that's not the official client and should be discussed with the third party developers. Ultimately, they will find out why the original Telegram doesn't do it.
4. Signal doesn't do cloud messages as Telegram does. You might have noticed that functionality like that makes Telegram much more popular among normal users.
On desktop* there is no e2ee even among the options.
* where "desktop" includes GNU/Linux phones.