I seem to be late to the party, but I would say, files are copyrightable, ideas (the tool they give details say for making) can be patented, unfortunately for the patent holder, this is exorbitantly expensive, and has a shelf life, of 20 years, where as copyrights are pretty cheap and are lifetime protection, or in some cases even longer. But the object, tool, chemistry, ect in question has to be able to be both useful and unique, so say patenting a new type of wrench, wont fly, instead you have to make sure to patent the new addition to the wrench, and not the whole thing, say a wrench for a fractal shaped bolt head, would mean patenting the fractal patterned head, and describe why it is better than existing bolt heads,
also, on the aforementioned, "they can just find another way to do said thing" this is why many corporations do what is called "bubble" patenting, where they try to think of every procedure to complete a given task, and then patent the tools for doing it. or their uniqueness to each other, and previous designs that might be similar. not cheap, but when applying that way, is cheaper than a separate patent for each component and its method of accomplishing the goal given.
If you have not contacted thingiverse though I would definitely do so, as they really don't care about disseminating ideas or anything, it is just a repository to give potential people interested in 3D printing, something to print with their printers, to get them to buy their first printer. and that is it, ie, marketing gimmick, so fighting a war with a patent holder is just not in their best interests. I doubt they would even bother to look and see if the claim is legitimate. I have heard of them just pulling things, to be "better safe than sorry" so to speak.
As for the prints not being as good, I think that if the files are accurate to the dimensions, then they can be used to generate CNC code as well as 3D printer code, its all just down to the software used to do interpretation, I have an all in one unit that does laser cutting, and engraving, CNC milling, and 3D printing, and all I have to do is use reversal of the shape, then do a subtraction from a solid block, so I have a cavity that is going to be "printed around" the air space that the part created, and flip it upside down, and I can then use that same code to cut away the material, as it will pace the cutter head around working its way down, thinking it is working its way up laying down material but instead it is removing it. So I would not count on everyone who uses those files just squirting plastic, some might cut it from metal and make a part, (or cut it from plastic or what ever it is intended to be made from)
So while your "do it first, faster and higher quality" is good, but proving you are the originator, have more ideas you are developing (and purchasing is supporting more great ideas) , is the best option. That and provide customer support, so people will go to you to get something that if there is an issue, or question, they know they can get those answers, and help, then keep track of your customers, so you are not offering help to people who bought from the counterfeiter is all, any one with any decency will not fault you and will support your right to refuse help to people who did not buy your product since you designed it and the counterfeiters did not ask, but instead stole your work.
If a very large company that you can pile up a ton of value to take from them in court and might be able to convince a judge to agree with you has done this, then you should be able to get a lawyer that will take your case just for the cut, thus eliminating the "dance around till they run out of money and temper" since they stand to lose a lot, and if they dont have any chance of winning, you can hurt them over a lot more because lawsuits over emotional trauma and the like when its a little guy against a behemoth, especially if things like stagnating your ability to be creative, can be argued and no company wants to go down that long dangerous and expensive road, unless they are pretty sure they can win. just some ideas and thoughts on your situation. good luck I hope you can solve it to your satisfaction.