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Eucalyptus cuttings do not usually root that easily; therefore, this method of propagation rarely works. However, the tree does propagate easily from seeds, which can be started directly in the ground or in a container and later transplanted.

The leaves may be turning brown due to a delayed reaction to cold damage. but it might also be a fungus. If the stems are still alive, the leaves will regrow without help if it is cold damage. In case it is a fungus, I would treat the tree with a fungicide. I personally like neem oil, as it kills any fungus and any pests that may be attacking the tree.

Eucalyptus roots tend to be shallow compared to other trees, going down about 6-10 feet normally. They are very water hungry and if they cannot find a way to get enough water in the soil, may invade pipes. You can try giving it extra waterings, to keep it from seeking out water from the pipes or redirect runoff water to the tree for the same effect.

Sap is not typically a big problem with these trees unless there is a pest problem or disease present.

It may be overwatered. Cut back on watering and check the roots for root rot. This article will help:
https://www.gardeningknowhow.com/plant-problems/disease/treating-root-rot-gardening-tips-for-housplants.htm

You can trim away any dead, brittle branches as these are dead. Other than that, anything that is still alive will regrow its leaves. Most of country experienced unusually cold weather this winter. If this is the case where you live, this was the likely cause for the leaf dieback on the tree.

It does sound like the eucalyptus has a fungus. Treat with a fungicide.

The tomato is underwatered. In containers, they should be watered once a day in temps under 85 degrees F and twice a day in temps over 85 degrees F.