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Just water the ground around the plant. Do not put water directly on the plant, its stem, or its crown, if you can help it.

Really, though. These plants should not need supplemented water, unless you live in a desert area. They are very, very drought tolerant, and can become very invasive when they get watered and fed.

In short, you shouldn't need to water. It will be the last plant to show any signs of drought. In fact, you can even wait until they start to go limp to water.

It probably is a combination shock of a new environment and overwatering. Let the top of the soil dry before watering again. Also, they prefer indirect, bright light so if it's right in the south window, move it back a bit. Also, this article says not to water directly into the center, which can cause rot. Try to water below the stems. Here is more:

https://www.gardeningknowhow.com/houseplants/wandering-jew/growing-wandering-jew-plants.htm

I don't know why it wouldn't. You will need to attach it to your trellis at regular points to train it upwards. If you are using a three-dimensional topiary form, you can train several stems upwards and continue pinching the ends for a bushy shape.

I found one chemical that has no effect on an Iris, but will likely kill Wandering Jew. This would be Sethoxydim.