Have a look at Asylee travel information (pdf), a fact sheet published by US Customs and Border Protection.
You should apply for a refugee travel document. Unfortunately, I have not seen anything indicating how long such an application may be expected to take. If your family emergency is such that you don’t have time for this, you should ask an immigration lawyer about the prospect for being readmitted if you have to leave before you can get a travel document.
If you do not plan to return to the US, however, you can just leave. To answer your questions:
Can I leave without withdrawn my asylum based on the information that I provided.
If you leave without the document, there is a risk that you will lose your status in the US.
Will I be allowed to leave the USA?
Yes. They won’t prevent you from leaving.
Is the DHS will ask me about my status or will not allow me to leave the USA?
No. They don’t much care about people leaving unless they are wanted for a crime or under court order not to leave. It’s getting back in that you should be concerned about, if you indeed do want to get back in.
To answer the main question first, no the US government will definitely not stop you leaving the US. Unless you are wanted for a crime (or subject to a court order) anyone is free to leave the US whenever they want. But there are other important issues.
When you seek asylum you are asking for protection from your own government. If after you are given asylum you use the passport given you by that government you are essentially cooperating with that government that you claim was oppressing you, leading the US to believe that you didn’t actually need that protection. If you leave the US for a long period, then that tells the US government that you don’t actually need the protection they are offering you, and they will likely withdraw it. You can mitigate the problem of using the passport by applying for, and waiting until you receive, a refugee travel document.
In short, you will be allowed to leave the US. But you will probably be considered to have abandoned your claim of asylum if you either use your former passport or stay away for a long time. If you want to return to the US in the future you will have to apply for a visa like everyone else. The abandoned claim may cause problems with a visa, since they may consider you applied for asylum under false pretenses. Have a look at the links in phoog’s answer for more details.
If you have any expectation of returning to the US in the future you should consult a lawyer with knowledge of the asylum process before you leave.
P.S. Have a look at the links in phoog’s excellent answer.
Credit:stackoverflow.com‘