and read "the limit of when tends to in is 1". It is just because of lazyness, or because the function is obviously not defined at the limit point, that we usually skip the " " part.
As for the uniform convergence of functions, you can use the "sequential characterization of limits" to prove everything you suggest: converges uniformly to as if, and only if, for any sequence converging to , the sequence converges uniformly to . Thus you reduce to a sequence and can apply the usual properties. For instance (under usual hypotheses, like continuous), for any sequence converging to , converges uniformly to , hence we have . And using the sequential characterization (in a different way), you deduce .