Ratings

The absolute maximum ratings in the data sheet prescribe IOL and IOH, but these ratings are also indicated under the DC characteristics. Why?

The absolute maximum ratings and the DC characteristics ratings mean two different things.

The absolute maximum ratings describe the conditions under which the device will not be destroyed or suffer permanent damage.

On the other hands, the DC characteristics are values for which the operation is guaranteed.

These two things are totally different.
Normally both are described, so there are no cases when only one or the other is listed.

The H level output current (pin 1) is -10mA for the absolute maximum rating and -1mA (MAX) for the DC characteristics.
I would like to use a value of -3mA with a supply voltage of 5V.
Would this be a problem?

Also, why are the MAX values for the absolute maximum ratings and DC characteristics so extremely far apart?

Using a value of -3mA with a supply voltage of 5 V would be a problem.
This operation is not guaranteed.

The difference between the absolute maximum ratings and the DC characteristics is due to the large variation between devices.

Levels that risk causing permanent damage (destruction) to the device and levels at which the device can actually be used (guarantee of characteristics) naturally differ.

The absolute maximum ratings are determined by several factors including the physical size of internal transistors and wiring.

On the other hand, the characteristics of semiconductor products widely differ according to manufacturing variations and usage conditions.
Manufacturing variations are particularly large.
Therefore, there is a big difference between the values that can be guaranteed as characteristics and the absolute maximum ratings.

Taking the example presented in your question, there is no guarantee that a current up to -10mA will flow.
As a device, the operation is guaranteed only up to -1mA.
Therefore, please use a current of less than -1mA.

The uPD780023 Data Sheet lists an operating supply voltage of 1.8V to 5.5V (cycle time 1.6μs or higher) and 4.0V to 5.5V when using an 8.38MHz resonator; but is the operation not guaranteed for a cycle time of 1.6μs or higher when using an 8.38MHz resonator at a supply voltage of 1.8V ?

The operation cannot be guaranteed.
The CPU's cycle time and oscillator characteristics are different characteristics, and the device's operation can be guaranteed only when both are satisfied.

In this example, why is the data sheet's supply voltage 2.0V to 5.5V with TDK-made 8.0MHz and 8.38MHz resonators?

These figures are oscillation characteristics provided by the manufacturer of the resonators, and do not mean that the operation of the microcontroller is guaranteed for up to this voltage.

The RC Oscillation Frequency Characteristics Table in the uPD789124 Data Sheet lists the following:

Also listed is the following:

When using 2MHz and 1.8V, doesn't the cycle time exceed the operation guaranteed range?

The high-speed mode cannot be used with 1.8V as you inquired.
In other words, use only the low-speed mode.
(As indicated in the Data Sheet, the specifications in the section on the oscillator standard indicate only the characteristics of the oscillator. They do not represent values for which the CPU operation is guaranteed.)