Another one is when one language uses a single word for two meanings which the other language has split. An example I cam across is when I translated the word "amber" meaning the orange color and the translation came out the other end as a word that purely means "fossilized sap"
I generally learn proper nouns first (following news is my biggest use of MT) and I've found it often helps to replace them by placeholders such as X and Y when translating.
As to sibling comment, IIRC Yandex translate has a nice feature where they'll provide a number of meanings for words when clicked upon in the source text.