Can my spouse's request for his legal fee payment not related to our marriage stop my request for no fault divorce?

Depends on the Judge but usually no.

Whether your finances are "joint" or not, if what you mean by that is accounts in both names, that is irrelevant. All assets and debts acquired during a marriage are community property unless you have a prenuptial agreement to the contrary. So if he incurred the legal fees prior to separation, they are technically a community debt. In addition all retirement etc. earned during the marriage is a community asset. So yes, he could well be given funds from some community asset to pay this community debt unless you can show that the debt was incurred not in furtherance of the community (e.g. he committed a crime and it's the fee for his lawyer).

His requests/demands cannot stop the divorce. But they can delay it significantly. Ask your attorney.

I believe it is called negotiation. If you want a no fault divorce, you might want to consider negotiating those fees with him, but put a condition on it that payment to be made after no-fault divorce is final. Doesn't have to be everything, but perhaps 25% to 50%, depending upon the negotiations.

I will need to have more of the facts. While the debts incurred during the marriage are normally marital debts, there are exceptions. If your husband incurred legal fees due to his own actions it is possible that you could ask to have an adjustment in any property settlement, including the allocation of debt. Regardless of the fact that you have no kids, no joint finances or property it certainly appears that you have some obligations which will have to be dealt with. You are risking your future by not having an attorney represent you, or at least look over any proposed settlement.

No, the fact he has other cases and other fees don't impinge on your right to seek a divorce and shouldn't affect how the case is treated in CA.