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Based on the appearance of the pods, the most likely explanation seems to be damage from insects or other pests feeding on the pods. The pest damage could then cause the gummosis and allow rot in. I am not sure of specific ways to control this unless you can find the pests that are responsible and identify them.

The gum exudate can be eaten, the taproot can be eaten and is described as “carrot-like” or “yam-like”, and the seeds can be ground up and used as a coffee substitute. The raw or roasted seeds have been eaten by aboriginal people and are high in protein, zinc, and magnesium.
I can't fine any information on the toxicity to dogs.
Check with your local vet for information on the safety of your pet consuming this tree.

https://www.gardeningknowhow.com/ornamental/trees/bottle-trees/growing-kurrajong-bottle-tree.htm

This is more than likely a reaction to the excess water. Many plants will exude this in the form of sap. There is not much to be done until the area dries a bit. This will return the tree to its normal state.

It is hard to know without seeing the damage, but it could be a few things depending on how it looks, and where is occurred first.

I would say, as long as the soil had time to dry out completely each day, that it is likely to have used up the nutrients in the soil. Especially if this is sandy soil, or clay. You may need to fertilize.

If the soil did not have time to dry out completely between watering, then it may have contracted an infection. Most trees that are able to store water, such as these, will overcome the infection on its own.

You can help it along by using DOLOMITIC LIME and WETTABLE SULFUR once per year, or if this happens in the future.

This article will give you information on the proper care of the trees: https://www.gardeningknowhow.com/ornamental/trees/bottle-trees/growing-kurrajong-bottle-tree.htm

The best time will be Spring, (Which we are at the tail end of here in the Norther hemisphere) however, Autumn can be acceptable as long as the weather is set to remain mild.

Here is an article for more information on the care of the tree:

https://www.gardeningknowhow.com/ornamental/trees/bottle-trees/growing-kurrajong-bottle-tree.htm

The foliage does not look like a KURRAJONG, which is a bottletree native to Australia. I would check out any native plant guides first for your area. You can often find those online and you may get lucky and find your plant. Are you sure it isn't one of the trees you removed coming back? They can easily return if all the root wasn't removed.