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Yes, this pruning will help rejuvenate an old tree. Just cut off any dead branches, keeping several live branches that will become your new crop. https://www.gardeningknowhow.com/edible/fruits/fegen/rejuvenating-old-fruit-trees.htm

Lichen and moss on a tree is generally not a huge issue, but combined with the loss of leaves, it may indicate problems.
Use the links below to help identify the issues with your tree.

http://msue.anr.msu.edu/news/yellowing_leaves_and_leaf_loss_reported_in_tart_cherries
https://www.gardeningknowhow.com/edible/fruits/cherry/cherry-tree-diseases.htm

https://www.gardeningknowhow.com/ornamental/fungus-lichen/treatment-tree-lichen.htm

Yes, the fruit will keep the seed slightly above the temperature needed to stratify these. You will need to stratify these on the top shelf of your refrigerator before planting. This will need a month, or two at the most.

It is helpful, but can be dangerous without gloves... If you remove that seed shell VERY CAREFULLY before stratifying, it will make germination time a few weeks rather than months.

Keep in mind that the seeds contain VERY high amounts of prussic acid (This breaks down into cyanide in the human body) and getting it on your skin can be fatal. PLEASE WEAR GLOVES if you are going to attempt to remove the shell.

It could be from water deficit - drought stress. Or it could be the result of a fungal or bacterial disease. You live on the south coast where, what country?

If there are "no leaves at all" what makes you think the tree is alive?
Reasons for defoliation and die-back can be water related, too much or too little, or a pest or disease issue, or an abiotic disorder.
In order to help you with diagnosis and treatment recommendations, we would need a photo and more information on growing conditions, your soil and water management program and pest control. What signs did you notice and when? Did it flower and leaf out in spring and then lose the leaves? Or did it not flower and leaf out at all this year? What changed this year? Was there any soil disturbance, construction, trenching, soil compaction, herbicide use in the vicinity, gas pipe leak, water deficit?

The most common cultivar is “Snow Fountains,” also known as “Snofozam” or “White Fountain.” The double weeping flowering cherry (Prunus subhirtella "Pendula Flora Plena") is sometimes considered a dwarf variety. This tree is usually 10 to 12 feet tall and wide, but some specimens can
This link has information on container Cherry Trees.

https://www.gardeningknowhow.com/edible/fruits/cherry/growing-cherries-in-a-pot.htm