No, I do not believe in miracles.
Miracles cannot violate the laws of science. If they did then you could measure the violation. The biggest violation would be of Relativity. A divine power that interacts with the world, even trivially, would create very different observations in the data than we have. In particular, conservation could not hold and as such would not be a binding constraint on the divine being. As a slack constraint it frees up a lot of other known processes. You could easily see such violations.
For miracles to occur they have to stay inside the rules, of course that rules out a divinity because a binding constraint prohibits divine interaction, otherwise its not binding.
Yes, the non-occurrence does undermine religion because religion is based on divine information which cannot exist under relativity, the physics precludes it if you follow the math to its logical conclusion.
If there can be no divine information nor divine intervention then a god could exist, but such a god would be close to the god of the deists. The difference would be that such a god could not give a divine plan, while the deist god could. Again, going back to the binding constraints problem, physics rules it out as a possibility. So a god could exist, but the god would be truly irrelevant and so its existence or non-existence cannot matter in any way.
Yes that undermines religion.