How do I connect text in Illustrator or InDesign?

To connect the shapes at the circles, try using the Pathfinder panels's unite function (first on the top left). Be sure to select the shapes you want to unite, first. OR, you can try using the Shape Builder tools to select what you want to connect, and then drag across from one shape to the next, to turn them into one shape.

As for the round edge rectangle, I assume you mean the bottom shape? If it's not behaving like something you can fill with white with a gray stroke, it's likely just a thick path, instead of a shape. Select it, then use the Object Menu>Expand, and expand the fill and stroke. That will make a shape of it. You may see your original path in the middle. You can get rid of this by selecting the new shape with the direct select tool, moving it, and then you'd see the (unstroked/unfilled) path that you can then delete with the regular selection tool. Or, just ignore that middle path if it doesn't interfere with anything.