As far as I know phpBB is capable to upload (attach) images out of the box. Why don't you want to use it?
I would like to create a separate page that will have the upload form for the images. I would like the page to automatically process what album to upload the image.
Use as target something like the following URL: upload.php?album=123&theme=my_theme
parameter determines the album, the theme
parameter let you use a different (light-weight) theme, if the default theme contains too much needless stuff.
I would like the page to then show the user the URL for the newly uploaded image.
By default Coppermine redirects to the newly uploaded image, but this could of course be changed. Alternatively you could use the Embed Code
plugin to display pre-built BBCodes / HTML embed code.
Could someone please do this for me?
Should this thread be moved to the paid support board? Sounds like you want to hire a freelancer.