I can have a 6-months J1 visa to work in the US and I'm wondering how I could stay working in the US after the visa expires.
On http://j1visa.state.gov/basics/common-questions/ it says:
My visa has expired during the program. What do I do?
If your visa has expired and you do not plan to travel outside of the U.S., you do not need to renew the visa.
So, is it legal to stay in the US after the visa expires?
Also, what about my girlfriend, could see come too?
The date of your visa's expiration has no bearing on whether you can remain in the country, or for how long. See https://travel.state.gov/content/visas/en/general/visa-expiration-date.html.
What matters is the stamp in your passport. If it has an "admit until" date, you must leave by that date. If it says "D/S," duration of status, you can stay as long as you maintain J-1 status. If you "complete" your program successfully, you get a 30-day grace period before you have to leave. Otherwise (if you leave the program early), you have to leave immediately.
Note that the question is asking about a visa that expires "during the program" -- because J-1 status is dependent on your remaining in the program. If the visa expires after the end of the program, the answer to the question is different.
The question you should be asking is not be whether it is legal to stay in the US after the visa expires, but rather
Is it legal to stay in the US after I finish my J-1 program?
And the answer to that question is
Yes, for 30 days only, unless you leave the program early, in which case you must leave the US immediately.
Some J-status nonimmigrants may be eligible to change into another nonimmigrant status without leaving the US. For more information, see https://www.uscis.gov/visit-united-states/change-my-nonimmigrant-status-category/change-my-nonimmigrant-status.
Your real question, though, is how to stay in the US working after your program finishes, so you need to find a nonimmigrant status that will allow you to do that. You also need to find an employer willing to sponsor you. This can be difficult, depending on your qualifications.
As to your girlfriend, she will only be able to come with you if she qualifies independently for a visa. If you want to sponsor her to come with you, you would need to marry her, and even then she would be only eligible to accompany you as a dependent in certain classifications. For example, if you had an H-1B visa, you could bring her as a family member in H-4 status.
According to the Common Questions you have 30 days to leave the country after your program is over. However, this implies that you may not work unless you have obtained an extension to your program. The answer that you quote seems to mean that if your visa expires but the original program is still continuing, then you can stay. Otherwise you have to leave within 30 days.
Can I participate in another J1 program after finishing my current program?
Your current J-1 visa is ONLY applicable for your current J-1 exchange program and under your current program sponsor. Upon completion of your current J-1 program, you are expected to depart the United States.
If you intend to pursue a 2nd J-1 exchange program (in a different category and with a different J-1 sponsor), you will need to apply for a new J-1 visa for your new exchange program and new J-1 sponsor – in order to reenter the U.S. for the 2nd program.
How long before my J-1 program starts can I arrive in the United States? How long can I remain after my program ends?
You may not arrive more than 30 days before the program start date shown on your DS-2019. Upon completion of your exchange program, you have a grace period of 30 days to depart the United States.
The remainder of the answer that you quote is relevant.
Your sponsor is responsible for assisting and advising you on all matters relating to your J-1 program, including authorizing travel outside of the U.S. and ensuring that your J-1 non-immigrant documents are valid at all times.
