Thanks all for the quick responses.
@firstdefence Thanks for the video. At first I was confused because when I followed your steps exactly, the context menu did not appear when right clicking. Using keyboard shortcuts for copy and paste also did not copy the selected area but the entire image.
However, this functionality changed after I tried the suggestion from @owenr to "rasterise" the image, even before trying to select an area of it. After doing that, the context menu appeared and copy and paste correctly copied the selected area. I find this a little odd since the image I was using should already have been a raster image rather than a vector, but apparently there is still a difference between "image" and "raster." This step was not necessary back in Fireworks. Rasterising a masked object makes sense to me, that was like the "Flatten Selection" command back in Fireworks.
@toltec Great suggestion and shortcuts. Ctrl+J is as close to as short an execution path as possible. This was similar to the Right Click selection > New Bitmap via Copy option in Fireworks. Will also keep Ctrl+Alt+Shift+N in mind, although that isn't the most convenient shortcut to use with one hand!
It's great that there are keyboard shortcuts but I strongly believe that all of these options should be added to the context menu so that they're easy for new users to intuitively find how to perform these tasks in the software without being specifically instructed. Right clicking a selection in Fireworks gave four very handy options: Crop Selected Bitmap, Crop Document, New Bitmap Via Copy and New Bitmap Via Cut. It would also be nice if you could also save a step by allowing the user to right click without having to switch back from the selection tool (marquee tool) to the cursor (move tool), since right clicking in selection mode doesn't seem to do anything. Ctrl-J actually doesn't require switching back to the cursor, which is at least good.
I think Affinity could really seek to improve in this area.
Being able to find how to use these functions without being specifically told is the mark of a well designed, intuitive application.
I'm new here and a bit sad about moving away from Fireworks but I will give Affinity a go. I'm not sure how responsive Affinity are at taking user suggestions on board but if they are, I would be happy to provide some feedback along the way to make the program more intuitive for new users and those switching across.