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Yes, a window box can be irrigated by a drip system, though it is not advisable to have a container without drainage.

Rain water is the most likely issue. If rain water can't drain freely from the container, root rot can kill plants quite quickly.
Here are some links to help you.

https://www.gardeningknowhow.com/special/containers/growing-plants-in-plastic-pots.htm
https://www.gardeningknowhow.com/special/containers

It's possible that the plants will suffer from too much water. I don't water my potted plants as long as there is water in the saucer. It's a good indicator, but not enough. Check the soil for moisture content with your fingers digging down a little, or with a soil moisture meter.

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-moisture-in-plants.htm

From your description I suspect "fungus gnats".
Here is an excellent article with several suggestions for control. You can determine what methods suit your situation and what materials are available in your area.

https://www.smallfootprintfamily.com/how-to-control-fungus-gnats-organically

The watering needed for plants depends on the type of plant, the soil and weather.

If you have more specifics we can try to help further.

Those are designed to filter out almost all chemicals. This will be fine as long as your fertilizer contains micronutrients like chlorine (which plants use just a tiny bit) but only a small amount is needed. So as long as you fertilize with a good quality food, then this will be very beneficial for the plants.

The type of water system is completely up to you and your local regulations. A simple drip line will suffice, and will cut the need for a trench as long as your planting spot has been has been prepared properly.

This collection of articles has just about everything you could need regarding irrigation, and I believe that you will find everything that you need here: https://www.gardeningknowhow.com/garden-how-to/watering