You are utterly and completely at its mercy. I.e. it has absolutely no reason to lie. If it wished, you'd be dead where you stand. The fact that you're even standing their proves it does not view itself as your adversary. It even acknowledges that your presence renders it and the cycles obsolete. Hence, you choosing a new solution.
And yes, Synthesis does solve the organic/synthetic problem. It is inherent in our nature to destroy ourselves/eachother. So we must change nature. Synthesis does this. We have common ground with synthesis. We understand eachother. We are now part of the same tree of life. We won't be killing eachother over our differences. If conflict arises in the future, it won't be simply because of the difference we had with synthetics or they to us.
In its mind it hasn't been exterminating anything. It's preserving the culture of entire advanced civilizations, giving the younger ones room to grow. If it didn't, the advanced civilizations and its synthetic creations would render the galaxy uninhabitable for future life. Which is what you're chancing when you choose Destroy. Same reason I prefer choosing it.
Considering its only a hologram there and doesnt actually have any kind of physical presence. Crucible could be preventing it from acting, or it cannot act apart from using Reapers and its not going to have a Reaper tearing through Citadel just to get to Shepard who is already half-dead. So instead of using the brute force approach it just used lies, half-truths and doubts to sway the only one person capable of to destroy it. After all if it did not give the destroy option it would have been so much more suspicions. No, this way with different options it makes the one option that it does not want Shepard to choose to seem less attractive.
And no it doesnt erase the problem. As I said the half-synthetic species that exist in the galaxy after the synthesis if they even have free will then all it did was change the "nature" of things. You can still create pure synthetics who are not bound by the change forced upon the rest. And there could be a species, an intelligent life that was not on a relay network, like the Mu relay it could have been blown away making the life on that planet completely organic if it was far enough.
Plus really the only way to stop conflict and make "everyone understand" is to force everyone to have the same values as you do, so basically it is a galaxy wide indoctrination anyway. And even if geth and edi become half-organic they are still incapable of reproducing the way humans or other previously purely organic species do.
So no, the potential for war, extinction and conflict is still there, especially considering that Reapers and their incredible technology would give anyone an edge over everyone else. The species that have fought in a war would not share the technology freely with species like Yagh for example or other ones who did not participate since that would create more potential adversaries. After all if a species did not know Reaper war, did not fight it and did not experience it, it has no reason to have the same values as the species that fought it.
Also IT DOES NOT MATTER, if Catalyst and reapers believe its the best thing to do, right now if for example the US government decided and truly believed that if it took control of the rest of the world by any means necessary, including genocide, nuclear bombardment and global war, it does not mean they are simply allowed to do whatever they want if they get defeated. It does no matter if they truly believed something, the crimes, the horrors unleashed need to be answered for. You do not let war criminals go, mass murderers or serial killers just because they truly believed that what they are doing is for the greater good.