Pasting images from clipboard in Safari
I know you can copy images in Firefox, as far as Chrome goes I haven't used it in years. With Safari however if you are talking about still images you should be able to Command-C an image. If that doesn't work you could always "Copy Image Location"/"Copy Image Address" in either Firefox or Safari, then open Preview and choose to open the URL you just copied. From there you can use the normal Select/Command-C/Command-V to paste whatever section of the image you'd like into any other system app that displays images (i.e. iMessage via the attach feature). That would at least eliminate the need for using Grab, however if you are okay with it you could also simply save the image itself temporarily, click the Attachment button, drag and drop the image from your downloads fan/folder and delete it when you're done.