There are a few simply checks to make for an ice production and small cube problem. There are only 2 things required for a refrigerator to produce ice.
The first thing is proper freezer temperature. The freezer must maintain 14 degrees or less. The second is a good water supply to the refrigerator.
If the temperature is good, we need to check the water supply. This can be done by locating the shut off valve for the supply water. Turn the water off before disconnecting the water line from the refrigerator. After disconnecting the water supply line for the refrigerator, turn the valve on and check the water flow from the supply line. There should be a good strong supply of water from the line. If there is not, the shut off valve on the supply line may have to be replaced. If there is good supply pressure, the problem may be in the refrigerator water valve or tubing. This will require a qualified service technician to resolve.
To address the bad taste from the water dispenser. There is only 2 things different from the water out of the refrigerator dispenser and regular water from the tap. The first and most common is the water supply line to the refrigerator, a rubberized supply line may give a water hose type of taste.
The other is that filters remove chlorination from the water giving it a different taste than from the tap water. The storage of non-chlorinated water can have an old or stale taste. I would suggest that you consult your owners manual to see if your refrigerator can be operated without a filter. If this is possible, remove the filter and flush a gallon or so a day for 3 or 4 days. This will allow the chlorinated water to clean the tubing inside the refrigerator. All the water lines in a refrigerator are made from food grade plastic tubing and do not retain any odor that creates a bad taste.