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Your temperature highs in summer match those of my area here in the united States. This is a very important factor when choosing when to plant something, but the other question will be... How cold will it get in your area? Likely, if it does not freeze at all in your area, then your best bet will be to plant anytime that it is not above about 30c.

If you do experience colder weather as well, then just make sure to give it at least 2 months to settle in before a frost.

This article will give you more information about these trees:https://www.gardeningknowhow.com/ornamental/trees/elm/camperdown-elm-tree-information.htm

Ragged, broken edges need to be pruned off. Look for the bark collar when deciding where to cut. If you lost large branches from the main trunk, elliptical-shaped cuts are made on the top and bottom. This is best left for a tree professional. Painting wounds is not recommended anymore. They cause more problems than they help. I have read that shellac is OK but don't use tar-based wound paint or house paint. Both are detrimental. Here are tips from the Arbor Day Foundation on storm-damaged trees.
https://www.arborday.org/media/stormrecovery/4_treefirstaid.cfm
https://www.arborday.org/trees/tips/keys-to-pruning.cfm

No pruning is required for this tree.

https://www.gardeningknowhow.com/ornamental/trees/elm/japanese-elm-tree-care.htm

These don't really need an extra fertilizer. You want them to grow a little more slowly to avoid contracting Elm Disease, which is a huge problem once it gets established.

If you still choose to fertilize them, you will want to choose a fertilizer you will want something with low nitrogen, and high phosphorus and potassium. You want this to be in slow release form, too.

Really, it is best to plant them in the correct soil, light, and area to get the most from the trees, as it is tedious to fertilize them without accidentally causing the death of the tree.

This article will give you more information on the care of the tree: https://www.gardeningknowhow.com/ornamental/trees/elm/camperdown-elm-tree-information.htm

Nope. I'm sorry, but that looks like a field Elm. Likely a start from a tree nearby. That is unfortunate.

Without a picture, and even then, it is hard to diagnose plant problems. If the droppings are very small and round, it may be caterpillar excrement. If it is more like sap or a constance oozing from the tree, slime flux or bacterial wetwood is possible. Contact an arborist or your local Extension Service for treatment advise. Here is an article on wetwood: https://www.gardeningknowhow.com/ornamental/trees/tgen/wetwood-bleeding-trees.htm. Here is the homepage for Texas' Extension; search for wetwood or slime flux. https://agrilifeextension.tamu.edu/browse/featured-solutions/gardening-landscaping/