These are really good questions, Martin. Thanks for posting them. I've never thought about how the way we perceive the meaning of these words could be language-related.
1) I see Right as something no one can take away from you (or no one should be allowed to, technically, but some countries don't worry about this too much), I think of basic human rights and rights given by law. They're something you can demand and should be enforceable.
Law is a field of study or an individual law (rule), either man-made or described by science (laws of physics).
Freedom and Liberty are ones I struggle with. I assumed they both meant general lack of control or oppression.
2) In Czech: right = právo, law (field of study) = právo or práva, law (individual rule) = zákon
so there is the same term for two of them but it's usually clear which one you mean.
Freedom and liberty would both be svoboda. Freedom of speech = svoboda slova, Statue of Liberty = Socha Svobody.
I had to look up the difference. If I'm correct, liberty is something everyone has but not everyone has freedom as it is more subjective?
We do have the word volnost as well. I guess it could be used as freedom but more in relation to animals or one's emotional state.
3) I am not sure but we might be able to see a pattern once more people have replied.