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It may be lacking enough water or it may have a fungus. Increase water to the tree and treat it with a fungicide to tackle both potential issues.

Yes, you can trim the tree to the desired shape, removing any unsightly or unruly limbs.

It may be a pollination issue. Particularly with loquats. They can self pollinate, but do not do it well. If you add to that a lack of pollinators, the tree will not produce. I would recommend trying to hand pollinate some of the flowers with a small paintbrush the next time it flowers. If all the flowers produce, it is a general pollination issue and you will either need to hand pollinate the tree or make adjustments to attract pollinators. If only some of the flowers fruit, it is a cross pollination issue and you may want to consider planting a second tree.

There are many reasons a tree can get yellow leaves. This article will cover some of those reasons: https://www.gardeningknowhow.com/plant-problems/environmental/plant-leaves-turn-yellow.htm That being said, tree diseases are very hard to diagnose without seeing the tree, so I would recommend hiring a local tree specialist to come look at the tree and advise you on it.

It may be a fertilizing issue, likely a nitrogen/phosphorus inbalance. Too much nitrogen in the soil will prodcue healthy growth but limite flowering and fruiting production. Try adding some more phosphorus-rich fertilizer or bone meal to the soil to sncourage more flowering and fruiting.

Neem oil is probably one of the safest products you could use. https://www.gardeningknowhow.com/problems/pests/neem-oil-uses.htm