I need to remove everything outside the black area and then combine what's left with the black. Could someone please give me a few pointers on how to do this?

First way

• Select all blue pieces and do path > combine.
• Optionnally duplicate black path (to further have a frame). Let's assume its a true black stroked path and not a black filled shape with two subpaths (inner and outer outline).*
• Select black (atop) and blue one and do path > intersection. This will make the cut in the middle of the stroke of the black path.
• If you want a frame, do path > stroke to path with the duplicate and select blue and black paths and do path > union.

* If the black path is actually a black filled shape, you'll need to select it and do path > break apart, and choose for the intersection wether you want the inner outline or the outer ouline to do intersection with blue path.

Second way : you could also select all and use shapebuilder tool to choose all the bits you want to keep.

Third way:

Group the black frame and the background.
Open the group, duplicate the frame and [right-click > Pop selection out of group].
[Path > Stroke to Path]  [Path > Break Apart]  [Path > Union]
Select the new shape and the group then [Object > Clip > Set Clip]

This lets you rearrange shapes later, if you have second thoughts.