What kind of things are impossible?
For example, paraphrasing quite a lot - uniqueness of names requires ordering; ordering probably requires consensus; consensus has a Sybil problem for which some kind of countermeasure is needed. So a namespacing problem that seems simple, and is simple in a centralised/trusted context, may only be practical to solve with a blockchain-like structure with a computationally-heavy proof-of-work in a distributed trust architecture. Even that may still be vulnerable to attacks from someone who can outcompute the rest of the network: and a Nation State Adversary (as a friend snarkily puts it) may actually have enough budget to try that. There may be another way, but maybe not another way that satisfies all the security requirements or that would survive an attack. And there are still other niches you need to worry about, like homoglyph attacks.
That's a long way to go just to make sure that there aren't two ~bobs! So you end up thinking, maybe it's better to find another way that ~alice knows she's talking to the right ~bob? Then you have rephrased your security problem as more of a UX problem: how do I try to avoid impersonation? - which is perhaps more practical to solve another way. [Edit: Also, now your users won't have to fight over who gets to take ~bob first. This may or may not be an advantage, depending on how you feel about username exclusivity.]
It's going to require a lot of hard work to solve this type of problem comprehensively; in the meantime, Signal does more or less the best we can do practically right now, and there's a lot of value in a practical solution that's best-in-class, works and millions of people can just pick up and use now.
Toc's original goal was to be a decentralized messaging app that makes no compromises in terms of UX compared to a centralized messaging app. Toc managed to tackle decentralized cross platform sync, but to this day I still have no idea how to approach automatic contact discovery and robust offline messaging.
To a point, so is offline messaging. Distributed datastores in general do that kind of thing - there are old implementations over Freenet in particular, and GNUnet as well and other things I think. The robust part is harder: if the client is offline, where's the disk space coming from? Volunteers' (nodes') cache? That needs to be opaque, and so do lookups. What's to stop spammers flooding the network? (Spam, and denial-of-service in general, is a Hard problem to deal with in general even in distributed systems.)
They're not among the hardest problems. Pseudonymity is a huge challenge as the latency gets lower against traffic correlation attacks, and a secure messenger is most definitely up against the kind of threat model that can and will try to pull those off in the wild. Perhaps mixing high and low-latency traffic may help, although it's going to need to be very carefully analysed how much (I believe I2P's design can do that, although I'm not sure if jrand0m ever implemented it in the router).
Thank you for all the interesting ideas and resources though! They'll definitely come in handy when I eventually revisit this space when I have more time later on.