Answer: Yes, this is exactly how it is done in the real world. What you have described is what I do in my job to check systems in CAD.
Since you indicated you would like me to step through my design process, I have detailed it below. Note that most of this does not involve CAD. CAD is invaluable, but only if you are prepared to get out the pencil and paper first. Also note that this is just my design process, it is by no means the only way to go about things.
When I begin any design, I start by figuring out general parameters like what space will it need to fit into, what it needs to interface with, what the input and output will be. Lets say (for sake of a concrete example) that I'm making a machine that takes in a long pice of pipe and cuts it into predetermined sections. My first steps would be to determine the size and material of pipe that I want to input, what lengths the machine should cut the pipe into, what method I'm going to use to cut the pipe, and how much space I have on the shop floor for said machine.
I would then proceed to make a really rough sketch of what I'm thinking. In the pipe cutter example, it might be as simple as a box labeled "cutter" and a line labeled with the pipe weight diameter and length.
The next step is to do some math to figure out what forces, speeds, etc will be required. Since you said you come from a mathematics background, I won't dwell on this too much.
I usually make another sketch here, except now I have some feel for how big my components will need to be because I know the forces involved.
After I do the math, I look for commercial parts that fulfill my needs and (if possible) download CAD files or mechanical drawings for those parts.
Once I have some commercial parts, I'll make another sketch, this time showing the relative positions of my commercial parts with their interfaces labeled so I know what kind of support structure I have to build.
At this point, I finally break out the CAD package. I start by making 3D models of whatever commercial parts I have that don't have 3D models available online and then proceed to my support structure parts and finlly assemble all the parts into an assembly.
Here's the part you are probably very curious about if you have never used CAD before.
In 3D CAD, you should draw each part (usually either something that comes as a commercial part or is made from a single piece of material in its own part file. Once you have part files, you can make what is called an assembly. An assembly allows you to select several part files and define the linkages between them. CAD programs have no sense of "collisions" so you have to tell the program which faces are aligned with other faces, what distance apart they are, etc. Each part starts out with six degrees of freedom and each constraint reduces the degrees of freedom by some amount. Specifying two faces are parallel removes one degree of freedom, a mate (two surfaces are coincident) removes two or three, etc.
I haven't used FreeCad, so I can't comment specifically on what the pros and cons will be, but my guess is this: the pros will be that it is free and you will see what dimensions work and what doesn't work, the cons will be that commercial software would have been easier to use.