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3DPrinting
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I can't comment on other software since my experience with the commercial options is near-zero. However, specifically in FreeCAD you can do some incredibly tricksy things with sweeps and lofts. I've gotten a lot of mileage out of them.
You could create your shape there almost entirely with lofts. A loft takes two or more sketches and you can think of them like "keyframes" in 3D space. Typically you would stack them on top of each other at the specific locations you need, and at the location of each sketch the 3D solid have a cross section of exactly the shape and dimensions of that sketch, and then the shape will be interpolated in the space in between in various ways you can select. The dimensions of each sketch can, of course, be completely parametric and as dimensionally accurate as you need them to be.
The pocket tool is also extremely powerful if wielded creatively. You can knock holes of arbitrary shape and complexity through things, not just circles and hexagons, to any depth. You can create complex three dimensional curves by making a solid of some shape or another, and then making a pocket all the way through it at right angles or indeed any angle. Think of it kind of like a milling machine pass.
Lofts and sweeps and pockets FTW! That does look like the right solution for single objects.
What about objects that sit at an angle relative to each other? Can you define workplanes (sketches) at arbitrary angles? Can you later slide those planes up/down, e.g. to add more distance from the face it would sit on? For instance, in the above object, the holders on the blue face are tilted 14 degrees from the blue base, and the little red holder in front is tilted an additional 5 degrees and lifted 6mm from the front of the larger red holder.
Yes, you can.
You can set up datum planes in arbitrary orientations but in all honesty, I've never bothered. You can position a sketch in any orientation, at any angle, at any position in 3D space. You are not confined to right angles of the X, Y, and Z axes.
Sketches used in a loft don't even have to be oriented on the same plane relative to each other.