Does the product a/c have a drain hole?

Yes, there is a drain, but not quite like you think.

Condensate collects in the outside housing of the air conditioner. It may run out directly (usually hole or notch in corner), or it may be retained for the fan to slap around and increase efficiency (helps cool condenser).

There are many reasons that an AC may not be able to go below a certain temperature. It could be by design, compressor failure, faulty thermostat, inadequate air flow through evaporator (inside grill/ condensor (outside coils), too large a room, and especially too much moisture in the air. Placing the AC unit in direct sun can impair performance, as can a windy breeze blowing into the condenser so that the hot air can't be blown out.

Yes, air conditioners can run all day long and throw ice cubes at you. But, if there is a lot of moisture in the air, the cooling effect of the air conditioner is spent on removing the latent heat in the air, and causing the moisture to condense. Only when the moisture level drops will the air start being cooled more effectively.

Air conditioners of your type have tiny passages that allow the water (if clean) to go to the back of the unit where a splash ring on the fan blade whips the water onto the hot condenser coil, eliminating the condensate outdoors and subcooling the refigerant in the condenser in doing so.
If you have water inside and its level or slight tilt back , then you need to clean it up or forget it!