Passport greencard thru asylum?

I don't think so. When you apply for asylum, you become countryless, which is giving up your citizenship. You should read this forum because they are discussing the same issue. A person who works at the immigration law office said:

"When reviewing your application for green card, the immigration officer must determine that you are still a refugee. If you renewed your passport, she can make a case that you no longer meet the refugee definition and thus not eligible for adjustment. Moreover, the law permits REVOCATION OF ASYLEES STATUS when you no longer fear persecution. The officer can then send your file to the asylum office for revocation proceedings or if you case was approved by a judge, she will ask the INS trial attorney to file a reopen motion with the judge to cancel the asylum grant."

Some people renewed their passports and got away with it, perhaps because the officer is human and she just does not want to make people's life more difficult than it already is.

But why take the chance?"

I would not risk and not renew my passport if I were you. Pay the fee and get the travel document each year until you become an American citizen.

That's between you and your original country. you remain a citizen of your country, so as far as the US is concerned you can still carry its passport.