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What does it mean expansion of space in big bang cosmology?

I am having real troubles wrapping my head around it?

Problem #1. They say (for example on Wikipedia) that space is expanding only on a very large scale but on small scale matter is gravitationally bound and therefore there is no expansion. In other words, if I understood it correctly, distance between clusters of galaxies is increasing but the size of galaxies themselves does not. Well, in the early Universe, right after big band when everything was more compact, all mater was gravitationally bound to each other, why was space expanding then?

Problem #2. How can we measure that distance between galaxies is increasing? It takes more time for a light to reach us?? Well, perhaps that can be explained by speed of light not being constant but rather slowing down as universe ages? So at time = 0 the speed of light = infinity and then it goes down as necessary to make the math work. I am not a mathematician but I think that if we drop the assumption that speed of light is constant and instead assume that the size of the universe is constant and on the large scale things do not go anywhere, the math can be made to work out. I think we can even explain red shift that way too. And if there are some minor problems with the theory we can always patch it by deploying such hacks as "dark matter", "dark energy" and etc., didn't we do the same with expanding space model?