What was the size of the heat pump you replaced?
Are the ducts noisy when both zones are calling for heat?
What are the sizes of the main supply trunk and return?

The air handler is available in a 3 or 4 ton size and is selected based on how much air it can provide. The 4 ton can work provided the duct work is capable of handling the air flow. The 3 ton handler may have also worked if it were a proper match. I don't think a single speed air handler will ever work properly with a zoning set up. I also question how effective a single stage heat pump will work with this set up. Perhaps others can comment on this.

I believe you have undersized ducts and a bad installation. Most HVAC contractors get into trouble when it comes to zoning. Hopefully this guy spends to the money to fix the problems correctly. I don't get a good feeling from what you have described so far.

How did you select this contractor?

The old heat pump is 3.5 tone too. I was told the duckwork is capable of handling 5 tone air flow. The zones worked fine with the old system too. The ducks are noisy whether one or two zones are calling for heat. The fan speed is the same. As I understand when one zone is calling the speed should be reduced.

I was told 3.5 tone outside unit can work with 4 tone inside unit with proper zoning. It looks like the issues are mostly due to zoning. They threw the old zone board out, and I don't even remember the make and model.

The outside unit makes a grinding noise from time to time. I was told that was normal. Really?