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The trees are southern magnolia (Magnolia grandiflora). They can be trimmed, but be conservative. Clean the interior canopy of dead branches, correct structural defects as best you know how, light thinning and shaping is okay. Extreme topping and side pruning that exposes the interior woody structure to sunburn would be detrimental.

https://www.gardeningknowhow.com/ornamental/trees/magnolia/southern-magnolia-trees.htm

It certainly is possible. Ask your condo association management to arrange an inspection. If the tree is in common area then the management should take responsibility for damage and repairs.

I am unable to zoom in on your photos to see details, but it appears that the tree may have 'magnolia scale', a sucking insect pest. If so, this may be a good time for treatment that targets the 'crawler' stage of the insect life cycle when they are vulnerable.

http://www.mortonarb.org/trees-plants/tree-and-plant-advice/help-pests/magnolia-scale-neolecanium-cornuparvum

http://msue.anr.msu.edu/news/check_magnolias_for_magnolia_scale_now

Those are seed pods and can be removed if they don't fall on their own. If you are concerned about losing more branches, consult with a certified arborist about pruning/thinning your magnolia. Mid summer to early fall is the best time to prune magnolias.

The poor appearance and leaf loss may or may not be due to the pruning. Excessive crown reduction and/or thinning, removal of more than 25% of the foliage, can be stressful and it takes time to recover.

You can use a 12-12-12 chemical fertilizer, but for the best for your tree, I would prefer if it were a high quality product like this, with organic materials and controlled release nutrients and a full range of major and minor nutrients.
http://www.sustane.com/products/nursery-greenhouse/sustane-12-12-12-90-day

Another organic all purpose fertilizer that is widely available: https://www.domyown.com/dr-earth-organic-all-purpose-fertilizer-p-9485.html