Certainly you need to fix what is broken as no lights and/or an exposed live fuse are both real and immediate dangers.
Can you post a photograph of the inside? (Carefully, as things might be live) That way we might be able to see if there is any obvious trouble brewing with your plan.
In principle it is straightforward to swap all of them out, the main thing is to make sure you work systematically don't mix up circuits when replacing them.
This is particularly important for the ring circuits (your 13A sockets) because if you cross-connect them it will "work" in the sense all sockets have power, but then you have a single double-length ring fed from two 32A breakers in parallel and that is dangerous as the fault current can exceed the cable rating by a large margin.
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Also worth knowing before you start is most DIN rail mounting accessories have the ability for the rail clip to be opened to make it practical to fit it on the rail from above (i.e. not sliding it on from the end). Here are some photos of a DIN rail fuse holder where you have to use a screwdriver to pull back the plastic moulded clip, and a MCB where you can easily slide the clip open or closed and it stays in place without any tricks.
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It probably is not the case in your unit, but some CU have a DIN rail with only half of the "lip" for holding the MCB, etc, and the other half is fixed by the connection in to a rigid bus-bar. In those cases unscrewing the bus-bar connection allows the device to be removed.