Welcome to the forum!
Often when new users say "layers", they're referring to objects which are stacked on top of each other, but they are really all in one layer. However, I get the sense from your message that you really are using layers, and that would explain why you can't move objects up and down through the z-order. I can't tell for sure if that's the problem, without seeing the SVG file. But I'll give you some brief info, and if it doesn't help, then please share your SVG file with us.
If there's some reason that you need each object in its own layer (like for example, you're planning to have the design printed on fabric) you can just rearrange the order of the layers. Open Layers menu > Layers. There you will see the layers listed in their correct order, and some green arrow buttons for moving the layers up or down through the list.
If you don't necessarily need the layers, you can move all the objects into one layer. Select it, then Shift + PgUp or PgDn to move to different layers. After that, switch to the Selection tool, if not already enabled. On the control bar (the first toolbar above the horizontal ruler) there are these 4 buttons:
,
,
,
. If you can see those tiny little arrows, you'll get the idea what they do.
So just select an object, and click this one
to move the selected object down by one step. You can click repeatedly to move several steps. Note that every object occupies it's own level in the z-order, whether they are on top of each other or not. Also note that if you make a Group, the group is considered to be one object, no matter how many objects might be in the group, as far as the z-order matters. It's possible to move objects in the z-order inside groups, but not between groups.
Last comment. If you get confused about what the z-order is, try Object menu > Objects dialog. It's a relatively new feature for Inkscape. Before we had it, you could only know about the z-order of objects by how they behave when you start moving objects up and down. But the Object dialog displays objects in their z-order. Hhmm....I haven't used it very much, but I think you can drag/drop objects in the Object dialog, to move them through the z-order. I think you could rearrange the layers that way too.
Let us know how it goes
