A: You should consult with an experienced immigration lawyer about the specifics of your situation, but as a general matter children of US citizen parents can acquire citizenship at birth even if they are born outside of the US. In order to claim their US citizenship, these children must apply for a Certificate of Citizenship. Whether or not you are eligible to do this will depend on the specific facts of your situation, and is something that can't be answered through brief online inquiries like this, which is why you should speak at greater length with an experienced immigration lawyer.
A: Peru is not a visa waiver country, so no matter what kind of status your character came to the US on she would need to apply for a visa. If she wanted to come as simply a tourist, she would need a B-2 visa, and she would be unable to work on that status. Her children would each also need to get the same visa to come with her. She would also be granted entry for six months with that visa, and that period could possibly be extended for a second six months, but then she would have to leave.
If she comes into the US on a B-2 visa, then later marries her boyfriend and files for a green card, she may face difficulties in the green card process.
If she wanted to come to marry her boyfriend, she
could get a K-1 fiancee visa. That visa requires that she marry her fiancee within 90 days of entering the US, and can allow her children to get derivative visas to accompany her. Once she's in the US and married to her boyfriend, she can request an adjustment of status to get a green card.
A: You need to meet with an immigration lawyer in your area who is experienced with asylum claims. From what you describe, it doesn't sound like your situation fits the requirements for asylum, but you need to speak with an attorney who can get the whole story from you to get an opinion on your chances of success in an asylum request. Asylum is a complex area of immigration law, and a brief summary of your situation like you've provided isn't enough to determine your chances.