How can I add paragraphs to a translated node?
If I remember this right this is by design (or maybe more a technical necessity). And you can only circumvent this by having multi-value fields inside a paragraph type. Like a node reference field for example. This then can have a different number of items per language and even in a different order.

You could also add a new untranslatable "Visibility" field to each paragraph type, like Paragraphs Collection is offering it, see 2846682#comment-11887724 for example. Then you add different paragraphs in the original language but set them to be displayed only in selected languages.

And regarding Paragraphs Asymmetric Translation Widgets you may file a bug report in its issue queue.