1. Since Telegram is freely available to the hunter-gatherer public of the worldwide internet, and the
Terms of Service 
one would have to agree to use it are very simple and do not seem to be in any conflict with Dhamma, as I understand it, I would think that allowance would be already given for the Sangha to use it.
But I can ask Mr. Durov on Telegram, as he seems to be reachable there publically. Not sure that I can reach him and make it clear enough. But I will try.
2. Since Telegram as it is would be already usable, if allowance is clearly given, it is not clear to me what special tool dedicated for the Sangha could/should then be built up. I am not sure what features/abilities such a Sangha messenger tool should have different from the common Telegram.
Just for an example: One could possibly build an own Telegram client which is restricted to only one channel/group (associated with the Sangha). But I am not sure that such would be a useful idea.
One could use Telegram authentication as a tool for registering and logging in to the forum (making use of the public Telegram servers to do so). There are many possible uses of existing Telegram features without a special own app.
3. Not possible to clearly answer without knowing what kind of tool exactly is to be built, and on what kind of platforms it should run. A web application, running on a website would be the easiest as a first step in any case, and then see how to adapt to mobile apps etc. which can be a bit complicated.
As for use of infrastructure: The existing Telegram server infrastructure with its public APIs would surely need to be used in any case for the app to operate. Since only the client applications are given as OpenSource to modify for own use cases to everyone, but the server side is secret and closed, and is surely very complex, not possible at least for someone like me to reverse-engineer and replicate, and most certainly requiring costly infrastructure for someone who would attempt to do so.
4. I don't understand what this reallyy means, and how it could be compatible with Telegram.
Maybe good if I try to invite Mr. Durov here to get in touch with the Sangha. He might be able to answer certain things more clearly.

https://core.telegram.org/api/terms
Open for companies to browse consumer, why not for communities of people. Somehow "first state in the new world".
* Johann : under current conditions, if having right view, my person wouldn't know why such would not get featured from above (one just needs to be clear that much needs to be explained).
Hmm, I had not seen the API terms of service before, which might be less liberal. Not sure.