Can an executor be paid to handle an estate if not stated in will?

You are in Long Beach, so I am assuming your aunt's probate will be in Los Angeles County, thus California probate law applies. Whoever is appointed by the Court as Executor, or as Administrator-with-Will annexed, will be entitled to statutory compensation. This is fixed by law for ordinary services, and is taken only with a court order from the probate court, and after the work is done. The estate representative will have to petition for approval of such fees. If additional work is done, over and beyond ordinary services, then application may be made for extraordinary fees.

Your cousin being named in the Will to be Executor, isn't automatically the person to select who shall do that work. He may be able to nominate some other cousin, but any of your aunt's heirs-at-law or her beneficiaries under her Will, may also seek appointment. This is done by petition to the court. The court will decide who is to serve as personal representative, based upon such petitions.

There are many fine probate lawyers in Long Beach; you will have no trouble finding someone to monitor the probate proceedings for you. If you think someone is seeking too much compensation, you could object to it. Meet with a probate lawyer near you, and the monitoring and advice you will receive should be worth the cost, especially if your expectancy share in the $1.5mm estate is significant.

The California Probate Code provides statutory compensation to an Administrator or Executor of a will under California Probate Code section 10800 (you can google that and read the actual statute). Compensation is based on a percentage of the assets: 4% of the first $100,000, 3% of the next $100,000, 2% of the next $800,000 and 1% on the next 9 million dollars. So your cousin is entitled to $28,000. Your cousin's attorney is also entitled to $28,000.

Many Administrators, especially when they are family, waive their fees but it doesn't sound like your cousin is going to do that.

Be aware that I was assuming that the Total Estate is $1.5 Million. The fee is based on the Total Estate before you subtract debts like home mortgages.