How do you stop tiny papercut details from tearing when you lift the piece off?

The stretching and tearing you're seeing with those thin membranes and spikes is almost certainly due to the pressure application rather than just the blade sharpness. When you cut paper, especially cardstock, you aren't just slicing; you are stressing the fibers as the blade forces its way through. If you press too hard, the material fatigues and tears along the path, even if the blade is brand new. Try lightening your hand pressure as much as possible and let the sharpness do the work.

Another trick for incredibly fine silhouettes like those wing veins is to use a micro-scoring method first. Instead of trying to cut the thin lines completely in one pass, gently score the lines with the blade until they are almost severed, then use a precision punch or a very fine needle to 'pry' the material away. This creates a clean separation along the line rather than a rough slice through the material.

Regarding the idea of blending the background first and then adding the black silhouette, that's totally doable and actually a common professional method for reducing waste and cutting complexity. You would apply the ink blend, allow it to dry fully, and then glue the pre-cut dragon onto the surface. You are right to wonder about the glue showing through the black card, though. If the card is opaque, a standard application of PVA or craft glue should be fine. Just make sure you use a very thin, even layer of adhesive, maybe even applying a spray adhesive lightly, to ensure coverage without creating pooling that could be visible. I'd recommend testing your silhouette pieces on a scrap piece of 160gsm before committing to the main project. Cut one of those tricky spikes, see how it performs, and adjust your technique based on that small sample. Good luck with the final touches on the piece for your dad.

I agree with the point about not pressing down too hard when cutting. I used to over-force it, which always led to the tearing you described. What helped me out the most last time I did somethig similar was adjusting my cutting motion. Instead of pushing the knife through with a forward stroke, I tried slicing the blade vertically through the paper. That motion prevented the material from bunching up at the entry point of the cut and kept the thin membrane pieces much cleaner. I found it made a huge difference on the fragile parts.

I totally get the worry about the thin bits tearing out, though I'm just starting to work on the final look myself. I've been experimenting with the black papercut dragon silhouette and trying to apply the ink blend, but the raw edges of the paper look pretty rough after the knife cuts through them. I'm wondering if there's a better way to seal those sharp edges before I start painting. I was planning on using a tiny detail brush, but I'm really afraid the black ink is going to soak into the fibers and make everything look messy. Does anyone have tips for getting a crisp line with the ink on heavy cardstock, or maybe a sealant I could use first?