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Weeping cherries are known to be short trees. Depending on the variety, they can be anywhere from 8 feet tall, to being able to top over 40 feet. And on the other end, there are dwarf varieties available now that are even smaller in stature.

What you describe is exactly what one would expect from the tree. You could even have to trim 3 or times per year when the conditions are right.

This article will help you to care for this tree: https://www.gardeningknowhow.com/ornamental/trees/weeping-cherry/weeping-cherry-growing-tips.htm

Water the weeping cherry a few times each week for the first few months after planting. Keep the soil moist to a depth of 1 to 1 1/2 feet. Check the depth of the moisture about an hour after watering. You can use a water moisture probe.
Water the tree once or twice each week after the first few weeks. It can tolerate dry soils after it has been planted for a few years but, for the first two to three years, the soil should not be allowed to dry out completely. Letting the soil dry will drought-stress the tree and probably kill it.

https://www.gardeningknowhow.com/ornamental/trees/weeping-cherry/weeping-cherry-growing-tips.htm

You can treat in spring, but it would be a good idea to treat now, also, with dormant oil. This can help stave off some of the infection and make it easier to treat in the spring.

The leaves should fall off by themselves as the new leaves start coming in, but if they don't on their own, then it can be fine to do so manually.

Some specimens can grow as tall as 20 feet. While some dwarves can remain as small as 3 to 5 feet.

Some specimens can grow as tall as 20 feet. While some dwarves can remain as small as 3 to 5 feet.

No, Weeping Cherry are likely grafted and the suckers would not likely be an ornamental tree.