Grout recommendation and how to get at corners?
That grout looks like it should work, the problem is that most grout require you to take out most of the previous grout. Certain groups, such as Mapei flexcolor, is built so you can add more to previous grout jobs, but it might have to be the same type.

As for the corners, you should use a colour matched caulking rather then grout. This allows for contraction/expansion. Most brands of grout have caulking in the same colours to make it easy to match.