To have transparency you need an "alpha channel", which is a fourth channel (besides the three R/G/B channels) that indicates how opaque is a pixel (like the Red channel tells how red it is). When you load an image from a JPG, Gimp assumes that you will export back to JPG (which is a format that doesn't support transparency) so it doesn't enable transparency by default on that initial layers (but other layers will have an alpha channel). But you can always add an alpha channel with Layer > Transparency > Add alpha channel
. You can spot layers without an alpha channel because their name is in boldface in the layers list:

To scale the selection, just use the Scale tool. However, when you scale a selection, the result is a temporary layer called "Floating selection". You can do further transforms (such as rotate or move), but eventually you have to do one of two things with it
- merge it back to its source layer:
Layer > Anchor layer
(or Ctrl-H)
- make it a plain layer:
Layer> To new layer
(or Ctrl-Shift-N). This is usually a better solution, because you can later do things (change colors, or move again, etc...) without having to redo selections.
There are also buttons in the layers list dialog for these actions:

When you have a Floating selection
up, many actions are disabled, so you have to do one of these before you continue editing the image.