It sounds like this could not be a more serious case, so you need to speak to your lawyer about these things right away. In general, with respect to your question, the court has no discretion to "strike" or otherwise dismiss a prior that is increasing your criminal history score and making you ineligible for safety valve. A mandatory minimum is just that: mandatory - regardless of cooperation or the nature of your prior convictions. It may be possible to negotiate a different charge in return for a guilty plea and/or cooperation. Again, you must speak with your attorney about the possibilities.
I am a CA layer and this is just meant to get you started in looking for the right lawyer! This is not legal advice. No Attorney/Client relationship has been formed. I may or may not be licensed in your jurisdiction. Please consult an attorney licensed in your jurisdiction for state specific advice.
Cooperating will help in persuading the USA to NOT pursue the mandatory rather than be of assistance to you at the time of sentencing should the mandatory otherwise apply. Note that for the safety valve to apply, you have to speak with law enforcement and fully disclose your knowledge of criminal activity but you will not be required to cooperate. Cooperation may be a term the governement insists upon in order for you to get them to agree not to pursue the mandatory but if you were otherwise eligible for the safety valve and your plea agreement does not require cooperation, you would not have to go beyond the disclosure of information.
You attorney can challege the consideration of your prior conviction by way of objecting to the presentence report. Whether your prior offense was by way of a no contest plea instead of a guilty plea will be of no consequence. What will matter as to the application of the mandatory is the nature of the crime for which you were convicted, more so than the sentence you received. The sentence you received will determine the look back period (it will be a 10 or 15 year look back period, depending on the sentence you received).
It is tough for us to offer more to you because you are not sharing with us the offense with which a mandatory may apply. It sounds like a drug mandatory and if it is a drug offense and the USA has included the proper drug weight in the Indictment AND they have filed the section 851 notice (of intent to seek life based upon two or more prior offenses), whether they can show the prior conviction is a qualfying conviction will be vital. If you do have two priors and if you are successful in arguing that one should not be considered, you'll still have a shorter mandatory with which to contend (and you'll not be eligible for the safety valve)
You have asked good questions and your attorney should be able to address them for you.