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It will be ok to start with, but yes, eventually you will need to fertilize it. Plants that are growing in pots have no way to reach out and get more nutrients from the surrounding soil and do not have the same process by which ground soil replace the nutrients that plants need. So they depend on you to replace them.

How often you fertilize, depends on the type of fertilizer (the package will recommend how often you should be using the product). For my own fruit trees in containers, I like to use a slow release fertilizer that is a little higher in phosphorous. That gets reapplied once every 3 months spring through fall and is not applied in winter.

No, there is no reason why you can't plant your veggies near a persimmon tree. Just be sure to allow enough space to ensure the tree's roots will not take up all the necessary nutrients that the vegetables require. Also, make sure the tree does not cast too much shade on the garden.

The tree may be stressed and if it is stressed, it will stop ripening its fruit. Try increasing water to the tree and that should help bring it out of its stressed state.

Something must have changed if the plants were doing well and are now struggling. Try to figure out what is different. Hopefully this link will help: http://treesandshrubs.about.com/od/fruitsnuts/p/GrowingJapanesePersimmon.htm

This article should answer some of your questions: http://www.agriculture.purdue.edu/agcom/newscolumns/archives/ITG/2007/August/070803ITG.htm

You can plant it now. The best time to plant fruit in Florida is during the rainy season, to take advantage of free moisture. Really, in Florida, you can plant fruit trees almost any time. This article has some info on persimmon: https://www.gardeningknowhow.com/edible/fruits/persimmon/growing-persimmon-trees.htm