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Staking is fine as a temporary measure, but the tree will eventually need to develop the trunk strength to stand on its own. When the tree blows in the wind, its trunk is induced to grow stronger.

Also make sure the stake is not directly touching the trunk. It should be a foot or so away, driven securely into the ground, and should be fastened firmly to the trunk with non-abrasive material such as bicycle tubes or an old garden hose. Two stakes can also be used on either side. Remove the stake either next spring or next summer.

You can also prune the long branches to take off some of the weight. This article has general advice on pruning willows:

https://www.gardeningknowhow.com/ornamental/trees/willow/how-to-prune-willow-trees.htm

As this article explains, applying compost and fertilizer can help willow trees grow:

https://www.gardeningknowhow.com/ornamental/trees/willow/grow-a-willow-tree.htm

This article explains more about fertilizing trees:

http://www.extension.umn.edu/garden/yard-garden/trees-shrubs/tree-fertilization-guide/

Watering the tree once a week could help, especially during dry periods. Water the whole area where the roots could be growing, not just right at the trunk, to get the most benefit.

Did the plants that died show any specific symptoms before they died, such as spots on the leaves, yellowing leaves, cracking or oozing trunks/stems? These could point to a disease problem or a nutrient deficiency in the site's soil. Another possibility is that salt or other toxic substances from a nearby road, or herbicides from a neighbor, could have entered the soil and killed the plants. This is especially a possibility if the site is very close to the road:

https://www.gardeningknowhow.com/plant-problems/environmental/salt-injury-to-plants.htm
https://www.gardeningknowhow.com/plant-problems/environmental/herbicide-plant-damage.htm

This article will guide you as for how far from the house a tree should be:
https://extension.missouri.edu/p/G6900

An arborist would be able to help you with pruning issues. I am not sure what kind of tree you have, but here are guides to pruning several types of weeping trees:
https://www.gardeningknowhow.com/ornamental/trees/weeping-cherry/pruning-weeping-cherry.htm
https://www.gardenguides.com/78263-prune-weeping-willow-trees.html

This is a very common problem. Beavers are also a problem in my area. So much so, that there is no limit on how many or when you can kill them. They encourage it, much like wild hogs in Texas. Now, I will say that willows will gladly sprout back from the base. IF you would like, you can prune this to a single trunk, or you can leave multiple trunks. Just know that, in the long run, this will be much harder to maintain. Fencing will help some, but beavers tunnel, and eat roots as well. It may be necessary to either trap, or kill them according to your local laws.

It sounds like your tree is experiencing transplant shock, which is very common. Here is more information about what to do:
https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm