I think a lot of the magic around summoning Voidsent is essentially space-time magic.

In order to bring a Voidsent into the Source, you either need to provide a vessel for them to occupy (i.e. what we did with our friends in our journey back from the source) or you need to open a voidgate or 'hollow' for them to enter directly (i.e. what we did during our last excursion to the Thirteenth). The same basic principles apply to any form of travel between the shards.

Reapers make pacts with Voidsent in order to bring them into the Source. The only difference is that instead of using an inanimate vessel (like a statue for a gargoyle) or a partially biological one (like the Diabolos Armament did with a partially biological Weapon made utilizing chimerical cells), the Reaper uses their own body as a vessel for the Voidsent. You could, in theory, do this with anything by using soul transfer, including ordinary people on other shards. I also wouldn't be surprised if the principle applies more generally to soul transfer, and there may be a degree of overlap here with what we saw with Athena's experiments, Ascian Primes, and even Aulus mal Asina's research in Garlemald. You essentially just have two souls bound simultaneously to the same bit of corporeal aether.

To answer your questions:
1) If you were to restore the Thirteenth completely to its original state, you probably wouldn't have any more Voidsent. But as we saw with the Empty on the First, even when you disperse the aspected aether it still may not be a change that we see completed in our lifetime. Alternatively, it may just be that there's a deeper reason why the Void needs to exist, and some people may choose to remain Voidsent voluntarily in the name of some higher purpose. Who knows.
2) If you're able to invite another soul into your own body (as Hephaistos did), then perhaps you could do the same for a Sin Eater. The main difference is that its relatively rare to engage with a Sin Eater that we can actually converse with. So the question would be whether you could form a 'pact' in the first place (assuming that this mattered).
3) The Sin Eaters are still around, as is the Empty. This is largely covered in the Eden questline. I think that if you go far enough into the future, the First might regain some semblance of the world that it used to be, which may mean that the Sin Eaters eventually regain their original forms as the aspected aether is dispersed.