Not sure about installing apps

Did you drag Firefox and Thunderbird from the mounted disk image onto the Dock? Because that way you start FF/TB from the disk image when you click the Dock icon. Thus the mounted disk image is still in use.
Or did you not eject the disk image, if you have installed the applications properly and dragged the icons from the Applications folder to the Dock?

Installing Applications in Mac OS X

To learn more about Mac OS X: Helpful Information for Any Mac User by GGJstudios​

 

Re-boot and then empty the trash.

 

Click on Finder and you will see FireFox in the bottom left corner with an Arrow click the arrow as it's an "eject" arrow and when it's gone you can remove the trash. All .dmg files when open show there and you always have to eject before you're able to empty the trash.

 

Install drag icon into Applications.
Dock drag from Applications onto the Dock.

 

I suspect you forgot to eject the disk images. It's a common thing to have happen for Mac newbies to the concept of a disk image - in order to delete the image file, it must not be in use anywhere. This includes having it mounted as a disk by double-clicking it - which means you must eject the virtual disk first, before you can throw away the image. This is simple enough to do - find the icon corresponding to the disk in Finder's sidebar, and click the little Eject icon.