Yes, a finite universe is a legitimate premise under the laws of General Relativity. But then, so are infinite ones.

Yes, the universe did have a beginning ( in all likelihood, anyway ), but that doesn't mean it needs to be finite.The Uni is 'FINITE' with a beginning.

I'm not sure what you mean by that. Such a universe would be closed, finite, smooth, and its mathematical description everywhere differentiable. It's one of the possibilities of the standard Lambda-CDM model. See here :What are the impacts of an absolutely finite universe

Shape of the universe - Wikipedia, the free encyclopedia