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If you purchase a bare-root tree, then of course plant during late dormancy (winter). A container grown tree can be planted anytime, aw long as you can keep up with water management while it gets established, but preferably in spring after flowering.

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

Most likely this wound will heal on it's own. Just keep an eye on it for pest damage and do not allow suckers to grow from it, simply snip them off if they start. Dressing or covering tree wounds can be more harmful in the long run than the wound itself. This article may help:

http://www.gardeningknowhow.com/ornamental/trees/tgen/wound-dressing-on-trees.htm

Make sure to keep the soil moist, but not totally wet, or dry for about a month. This will allow time for proper rooting. You might even have a soil infection that needs to be addressed. Most can be cured with dolomitic lime and wettable sulfur. It is also recommended for prevention as well. You can add this two or three times per year to the soil to prevent infection, and condition the soil.

This article may help you with your weeping cherry problem

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

Not usually. They may have some of the fruit, but they will not usually hurt the tree. This article will help you with keeping birds out of the area: https://www.gardeningknowhow.com/edible/fruits/fegen/protect-fruit-tree-birds.htm

Trimming might help quite a bit, but you should expect a long recovery time before it returns to its weeping stature. Try this: https://www.gardeningknowhow.com/plant-problems/pests/insects/organic-snail-control.htm