Other Tox clients can work in those conditions, but reconnecting isn't very fast (at least for me). It might be a problem with Antox. You could post your issue here and see if they fix it: https://github.com/Antox/Antox/issues
Antox works well from what I have heard. Your issue sounds like a problem with internet connection. You get disconnected from the network, so it shows your friends as offline (because you can't connect to them). It might also have something to do with this bug: https://github.com/Antox/Antox/issues/477. You can try TRIfA too, but it's not an official client.
Well there is another problem in such case. As I understand mobile app is going to consume battery and network bandwidth a lot since there is no support for push notification. Some details: https://github.com/Antox/Antox/issues/6 https://github.com/irungentoo/toxcore/issues/1251
Yeah, but couldn't that be equally fixed by having homeservers? Note that i see at least two Antidote and Antox, dunno how well they work.
How do the matrix servers know each other? Basically the Tox DHT can search up anyone by the public key.
Ideally, you have both, i.e. programs can easily switch between via-homeserver and direct mode.
Btw optimally i think communication systems just got an address and some instructions like desired level of security/one/both ways, and it'd figure it out. Basically some programmer makes chat programs, and says "fast, one way" or "latency <50ms, one way, high privacy", a game "low latency, both ways".. The comm system knows about programs for communication and figures out which, with some way to configure it. But i am not sure how achievable that is :)