FYI, and IANAL, this isn't legal advice, etc. etc.
Patents are used to restrict the use of something, while in the process, telling exactly what that thing is, and how it works (since, how else could someone determine if they've violated it?).
Patents can be used in a de facto free manner (see: Tesla indicating their charger patents shall be free to use), but it seems like added cost for no benefit here.
An open design, of a manufactured part, particularly something with large barriers to entry -- for example, are those plastic parts molded? -- doesn't seem very useful for either party (mfg or customer), as it exposes your own product to copying, and your brand to dilution and counterfeiting (substandard parts being represented as, or confused with, your own). Patenting can be used to protect the mechanical design itself, independent of trademarking, which helps with securing profits (you hold a monopoly over that design).
Keep in mind that patent litigation, these days, is immensely costly, largely used only as a cudgel by large corporations to bully around each other, or smaller corporations. The cost of the patent itself is largely irrelevant (some $thousands for the paperwork, some $10...100k's for lawyers/artists to draw up?) while the cost of litigation dominates (millions). Which can be used judiciously (actually protecting the thing), rent-seeking (extorting settlements out of potential violators; see also: copyright trolls), or strategically (the case isn't actually meritorious or intended to win, but conducting it just to bully the opponent with $M's of their own legal fees to run them down).
Note that patent and trademark are separate mechanisms. You can still own the trademark, and prosecute violators of that mark -- whether or not the item it's on is patented.
Likewise, copyright is separate, and serves to protect an artwork rather than a method. The product documentation would be copywritten, while the product might be patented. Both might contain trademarks.
3D models/prints, I think are protected by copyright (would be something like sculpture law?), and needn't be patented necessarily, but mechanical things could be. Note that a copywritten item could be modified (fair use) or imitated (an original work of similar nature, but not infringing) for the same functional purpose; whereas a patented item can't be modified or imitated for the same functional purpose (that function is protected), but can be modified or imitated for a substantially different purpose (while having largely the same apparent form). I don't know offhand how exclusive these two roles are; presumably, an item could have both, and thus be protected from both kinds of reproduction.
As for publishing: having it online somewhere, as a whole design, with accessible formats (preferably industry standard interchange, e.g. PCB gerbers and (PDF?) schematics, 3D STEP or STL, etc.), is enough. It can be on your own website, or any other place that hosts such (indeed, Github is a popular choice).
Like, I have a couple such things on my website; though small, and few of them (and I don't have an index enumerating them, so you're welcome to try and find them
). Almost all my content is under default copyright conditions (I haven't registered anything; US law provides for an implicit creator copyright, and for which I can file a (retroactive?) registration if I need to file a lawsuit protecting something).
Tim