The milky sap is irritating so you don't want to get it on your skin. It turns black and is sticky so it isn't all that easy to wash off. You will need to decrease any tools you use or they will get gummed up too.
Once the sap dries or stops dripping from the branches it is safe to be near the plant. When I prune my plumeria, I have to do it from the inside out or it will rain sap on me.
Plumeria will lose their leaves naturally so most of the time I don't cut off the leaves. If the leaves are clean and the sap is dry it does not hurt other plants. The leaves will get very crispy and crumble when they are dried.
I try to rake up and dispose of most of the leaves I find because of plumeria rust and fungal diseases which my plumeria gets every year. Sanitation is my best control since I am not going to replace the trees with resistant cultivars yet. My trees are at least 35 years old and came with the house. There is a systemic fungicide I can use but I can't find it here, it is called fung-away. It is a grass fertilizer, but apparently I read that it works on the plumeria rust.
Happy gardening in Hawaii. Gardens are where people grow.