I’ve been using Inkscape for a while and I have no idea what he’s talking about with breaking curves apart.
If you want to add an eye, you need to “difference” not “union”. Union adds to the design, so it will just become part of the larger piece and fill in. Difference will cut it out of the larger design and leave a hole.
The letters will not move because you “unioned” them and they are part of the whole piece now. You have to go to Path and “break apart” to separate them again. They will all be separate things, so you will have to “group” them to keep the letters together as one object. It will also break apart all of the other separate pieces of the design, so you will need to select them all and “combine” them to put it all back together.
It’s a lot of work to fix something this way, but if you are still editing it, you can just undo the union of the letters and center them with the “align and distribute” tools. If you can’t undo the union, it might be easier to just draw a box over the area with the letters and “difference” it away and start again.