Hello and thank you for entrusting me to answer your question. I am very sorry to hear about this unfortunate situation.
While it certainly appears as though you are the victim of company politics, I must tell you that the conduct you describe is not ordinarily illegal, nor would it form the basis for a wrongful termination claim.
Generally speaking, employers enjoy tremendous discretion with regard to how they choose to manage their business. There is no requirement of fairness in the workplace, and supervisory decisions can be arbitratry or just plain wrong without running afoul of the law.
A claim for wrongful termination would arise only if you could demonstrate that the basis for you being pushed out was a "protected trait," such as your race, religion, or gender, in which case you would have a viable claim for discrimination. Otherwise, the conduct that you describe, while certainly unfair and and unforutnate, does not violate the law.
With regard to unemployment benefits, an individual will be eligible provided that they have received enough wages during the base period to establish a claim (either $1300 in one quarter of their "base period," or at least $900 in their highest quarter and total base period earnings of 1.25 times their high quarter earnings), they are physically able and available to immediately accept work, actively seeking work, and unemployed through no fault of their own.
Most often at issue is the final requirement--that the claimant be unemployed "through no fault of their own." This requirement will exclude employees who quit or resign, since the employee has made the conscious and voluntary decision to become unemployed. This requirement will also exclude employees who are fired for "misconduct," which is generally defined as conduct that evidences a willful disregard for the employer's interests (e.g. showing up to work drunk or stealing).
Poor performance ordinarily does not qualify as "misconduct" pursuant to the EDD's definition. If you are fired simply for "not cutting it," or even for making some unintentional mistakes, that almost certainly will not rise to the level of "misconduct," and assuming you otherwise satisfied the requirements for benefits, your claim would be accepted.
So, to answer your second question directly, you would actually NEED to decline your company's request for your resignation if you wished to remain eligible for benefits. By voluntarily resigning, the EDD will reject your claim for unemployment because they will find you to be unemployed "through fault of your own."
If you wish to remain eligible for benefits, you must not be the "moving party" in the separation of employment. You must make your employer force you out.
I recognize that this may create more tension than going with your employer's suggestion of resigning with the hope of being reabsorbed somewhere else in the company, but this is something you must overcome if you wish to ensure your eligibility for benefits.
Please do not hesitate to let me know if you have any questions or concerns regarding the above and I will be more than happy to assist you further.
If you do not require any further assistance, I would be most grateful if you would remember to provide my service a positive rating, as this is the only way I will receive credit for assisting you.
Finally, please bear in mind that none of the above constitutes legal advice nor is any attorney client relationship created between us.
Very best wishes to you.