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If it is a black walnut, than no, I would not recommend planting them near the tree. But if it is another variety of walnut, this should be fine as only black walnut is allelopathic.

Sap flow does not hurt the tree. Some trees have free-flowing sap that “bleeds” after late winter or early spring pruning. Pruning in early to midsummer, after the leaves have matured, will prevent unsightly sap flow.

It is true if it is a black walnut tree. Black walnuts are poisonous to most plants. This article will help: https://www.gardeningknowhow.com/edible/nut-trees/black-walnut/black-walnut-compatible-plants.htm

It sounds like it may have walnut blister mite. This article will help:
http://ceventura.ucdavis.edu/Gardening/Coastal/Veg-Fruit/PVD_Menu/Blister_Mite/

The black fungus may be attracted to the weakened tree. There are a few kinds it could be, but they can all be treated with fungicide.

To tell the truth, I would recommend treating the tree with neem oil. It is both a fungicide and a miticide and is safe for people and pets. Here is more information on it: https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

Most pine trees do not grow well within 50 feet of black walnut trees. If the black walnuts are closer than 50 feet to your pine trees, we recommend that you move them to a better location.

This could have happened for a few reasons.

First, this tree has both male and female flowers. If the male flowers did not bloom well at the same time as the female flowers, then you would have reduced or even no fruiting.

A late spring cold snap may also be to blame. The flowers are far more tender than the leaves and branches, so while the tree itself would ahve looked fine, if you got frost after the flowers opened, that would ahve killed them and then you would have no fruit.

Another possibility is that the tree has passed its fruiting maturity. Trees, like animals, have a life span and a set amount of time it can produce. It is possible that your tree has just gotten too old to produce. But, normally, if this is the case, you would have seen a gradual decline in production over the past few years.

Poor conditions may also have been to blame. High heat and humidity during pollination can make the pollen sticky and clump up, which makes it difficult for the pollen to get from the male flowers to the female flowers.