From my point of view there are two misconceptions in your post.
1) you need to be set up to work-for-hire to write an invoice.
2) you need to get clearance from their employer for things outside work hours.
ad 1) No, you don't need to. At least in Germany anybody who's legally competent can write invoices. If the invoices are secondary income, you will be taxed heavily (and declare it you must), but that's it. It has been some time since I last lived and worked in the USA, but I mean to recall that it was basically the same. Of course, invoiced money is your money now and you need to donate it to the FOSS project, which then needs some kind of treasury. But you said as much already.
ad 2) No, you don't need to. Your employer is your employer, not your owner. Now I don't know about the USA today (see above) but in European countries what you do outside working hours is your private affair -- discounting a few, very specific fringe cases. If you play soccer, dabble in explosives, or code for money doesn't matter. And frankly, your typical employer in most cases does not care anyway.
1) That may be allowed in Germany. Definitely not in Poland and many other countries.
2) In my experience, not true. Most often an employee needs to get a pre-approval that often take too long. As a full time developer, there's difference between playing soccer and developing software.
In my country a lot of people in IT are contractors (not employees) and sometimes these contracts are wild (like not working on anything else during that time and stuff like that).
That kind of clause doesn’t fly in either the UK or US, since it is disguised employment. The definition of a contractor is someone who sets their own rate and hours, and works under their own direction.
ad 1) No, you don't need to. At least in Germany anybody who's legally competent can write invoices. If the invoices are secondary income, you will be taxed heavily (and declare it you must), but that's it. It has been some time since I last lived and worked in the USA, but I mean to recall that it was basically the same. Of course, invoiced money is your money now and you need to donate it to the FOSS project, which then needs some kind of treasury. But you said as much already.
ad 2) No, you don't need to. Your employer is your employer, not your owner. Now I don't know about the USA today (see above) but in European countries what you do outside working hours is your private affair -- discounting a few, very specific fringe cases. If you play soccer, dabble in explosives, or code for money doesn't matter. And frankly, your typical employer in most cases does not care anyway.