A search for some guidance on how to do this led me to an old thread, but instead of playing necro I'll ask and try answer, since this is my first ever post.
1. I have dropped an image with transparency onto a background, but I want to remove the sharp edges it leaves behind.
I effectively want to feather the image in. One way to do this really easily is following:
I assume you already merged the images into one, So add a new layer, and make the new later the active one.
Take the paintbrush tool, make the tip about 8 pixels wide (less if your image is smallish), and draw in any colour a nice outline around the new object. Take the brush over the background but catch the edge of the overlay as you go.
Select the magic tool , and click anywhere on the outline you just drew to make a mask.
Deselect the new layer.
You will now see your mask over the original image, now select the blur tool (I used Blurs/Gausian blur)
Play with the amount of blur, but 2 pixels is normally going to do the trick.
I'm pretty sure this trick is common knowledge, but my google-fu was just not cutting it. This obviously works best if you have a solid background colour, if your background is more complex, then smaller brushes and fine control are required, but this should take you all of 60 seconds.