Lots of things: vote for their government officials, express their opinions about anything they want, travel anywhere in the country without special papers, work wherever they can find a job, marry whoever they want to (pretty much) ...
Yes, if they are born in the US, they are US citizens.
No.
Yes, US citizens can travel to Australia.
Yes, US citizens can travel to Japan.
Yes, US citizens can travel to Singapore.
Yes, US citizens can travel to Uzbekistan.
Yes, US citizens can travel to the Philippines.
it has set laws in it that protect citizens
The US classifies non-citizens based upon the avenue of entry into the US.
No, they are Mexican citizens.
The US flag symbolizes freedom and liberty that US citizens have. The US flag symbolizes freedom and liberty that US citizens have.
Yes, US citizens can travel to New Zealand.