If the company culture doesn't suit you, that's o.k. To me, the bigger cultural disconnect is over going to lunch because it relates to social norms. The second take home is more or less a confirmation of your disconnect and that's why it is framed in terms of ethics and professionalism.
People struggle with social norms when it comes to remote work. It takes extra effort to treat remote workers well and so it is not uncommon for people to take a pass on it.
Good luck.
This is an interesting thing they are doing to you. It is like they are testing you, but it can be seen as them taking advantage of you.
Maybe they are testing you to see if you can handle it, but most companies would give you little meaningless projects, but to have you work on company projects seems very odd.
I had to go through 3 months of intensive training AFTER I was hired before my former boss would even let me code within his beloved software program.
I know some companies, like Google and Facebook, spend hours interviewing.. initial phone interviews, flying people out to the headquarters, and giving them various tests, so its not unreasonable to know that they are trying to test your skills to see if you are a perfect match, especially if they are giving you the opportunity to work remotely. They are seeing if you will actually do the projects and complete them in a timely manner. Trusting someone to work from home and complete projects on time is a big deal.
If you are worried, try and research the company on www.glassdoor.com.
You could also just bluntly ask if they are testing you with intent to hire or compensate you for the time you have already given them.
If they tell you not to bother coming back, at least you won't be wasting anymore time there.