Best to show your criminal and immigration papers to an experienced immigration attorney before filing for naturalization.
Mr. Shusterman's (former INS Trial Attorney, 1976-82) response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.
To evaluate these issues properly, it would be necessary to review the facts, underlying immigration court history and documentation as well as your criminal charges and disposition sheet. You are urged to consult an experienced immigration attorney before taking any steps towards naturalization.
Short answer is yes. Once you are granted cancellation of removal your slate is wiped clean and you can travel and become naturalized. You will be issued another green card if your old one is taken away but your residency date will not change. We had a similar case for simple possession of marijuana and after winning his cancellation we immediately filed for his citizenship which was granted.