The black sooty mould is a secondary thing really, what's happening there is that aphids are feeding on the leaves - they secrete honeydew (shiny and sticky) and the sooty mould, which is a fungus, feeds on the honeydew. Get rid of the aphids and you won't get the sooty mould, I spray with jets of water in early may as soon as I see signs of aphids, this drastically reduces the population.
The most frequent problems I have had with plum trees have been due to fungal attacks of one sort or another. You comment that the tree was heavily laden last year - maybe a clue there. Fungal growths enjoy conditions of poor air circulation, so maybe thinning out some branches would help too. Your fruit may well be going mouldy because they are in these conditions, damp and with poor air circulation. Again, thinning out branches and removing some fruit will help. The orange goo I'm unsure of, it could be just the fruit rotting after fungal attack, or could be due to wasps, once they've made holes in the fruit, the juice oozes out.