Some good questions here. Though I am no expert I'll try to answer them as best I can.
Let's start out with the more general question: Why should we be encouraging manufacturing jobs to stay in the United States in the first place? Is there something specific about manufacturing you think we need to protect, or do you just dislike ANY jobs leaving the country? Manufacturing goes overseas because it is more profitable to do it overseas. It's just not something the US has a competitive advantage at doing. Our competitive advantage lies in other things - great IT, great emerging technologies, great business practices, great universities, etc. There is no need to specifically protect manufacturing jobs. If, on the other hand, we're talking about protecting jobs in general, there are far easier ways to do it than this. I'm not sure this would even achieve the desired goal of higher employment.
In order to answer this I must start out by asking an economic question. What creates wealth for a country? One of the first things that most economists will say is "products". The best and only way to get a product is to make a product. This is what manufacturing does. Creates products en masse. The more of a product that you have the more that you will sell. (assuming of course that the product is something that people want) The more wealth that will be brought in.
The next thing economists will say is bringing in money from outside the country. This also is what manufacturing can do. Only difference is selling the product outside the country. This is why exporting goods is so important to any country. In fact that is one of the biggest reasons that China has been able to catch up to the US so quickly in terms of wealth. They increased thier production of goods by creating and bringing in factories from other countries.
So no it is not just about the jobs. Though that is certainly another major factor for this idea. The wealthier a population the more money will be circulated, which helps keep any economy going. The only way to make the population wealthier is by keeping the population working and getting paid. One of the biggest reasons for a recession is people stop buying things. If people have money, and know that they will continue to make money, the more they will be willing to spend.
Now, onto the more specific question, about government subsidies for R&D and protecting those developments. I think we need to examine why the government subsidizes R&D for certain technologies in the first place, and just eliminate it if there isn't a good reason. If it's outright corporate welfare, then we would be better off just ending the subsidies instead of implementing a complex regulatory framework to protect the innovations. On the other hand, if there is some compelling public interest in developing certain technologies, then it really shouldn't matter where they are manufactured, as long as Americans have access to them. In either case, I'd be against a mandate to keep them in country.
For this there are many reasons to support R&D. A major one that should concern everyone in the country would be that if they don't then it is quite possible that other countries that are willing to give out money for R&D will eventually surpass the US technology wise. (for the simple fact that the more money being spent on it means more resources are able to be dedicated to what ever is being R&D'd) Quite possibly to the point were we would be like N.Korea vs the US right now. All bluster and not a big enough stick to actually threaten anyone. For this reason it is extremely important for the government to continue to help fund R&D. Now I'm not saying that this is an IMMEDIATE concern. But looking in the long range goals it certainly is.
Furthermore, I think the idea is unenforceable. Would they be required to manufacture ALL of the product in the US, or just the product that they were selling to Americans? What if they also received R&D subsidies from a foreign government? What constitutes "government assistance" to develop the product - I'm sure the definition of that would be litigated to death by every company that it affected. What's to stop a competitor from manufacturing the product elsewhere, thus eating into the creator's profits because they can't compete on a level playing field?
If they recieved nothing but US subsidies then yes they would be required to manufacture all of the product in the US. It would be pointless otherwise.
I'm not really sure about what to do if the corporation recieves subsidies from other governments. Perhaps in this case we could say to go ahead and allow them to use factories in other countries...however they would have to create an equal amount of factories/jobs in the US? Put it on a 1:1 basis. Though I am unsure of how this would affect prices... Ultimately the questions that you have in this paragraph would have to be answered by people far smarter than me.
As for what constitutes government asssistance? We can keep it simple to start out. Any company that recieves monies/subsidies from any government agency would be subject to my idea.
As for whats to stop a competitor? Copy right infringments. Pretty much any company will put a copy right on any product...often in many different countries. And they will do whatever they can to keep that copy right as long as possible. Since my idea only allows for a company to have to make the product in the US for a certain amount of time they will eventually be able to move the manufacturing over seas. Between copy rights and them being able to eventually move to "greener" pastures they should be protected in this way.
Generally speaking, most people benefit from free trade, not protectionism. If the government is funding R&D for no compelling reason, it should simply stop. And if it has a good reason, then it should celebrate the fact that the product might be manufactured elsewhere under cheaper conditions, thus increasing its availability.
Part of this was answered above so I'll only address what I haven't addressed already. You are correct that generally most people benefit from free trade. But as with anything else...too much of something can be a bad thing. There has to be a middle ground. As it stands now there are too many corporations which have thier main officies based in the US (partly because of the subsidies they are able to get by being based here) but thier manufacturing jobs are outsourced to outside countries. This brings in no wealth for the country. In fact it costs us in the long run. This is one of the reasons that we are having such a hard time in this country right now. Too much money going out...not enough coming in. Partly due to the Free Trade Agreement.
Hope I satisfactorily answered at least most of your questions.
