Grammar isn't really my strong point but I believe the only time you don't need a preposition after 'antes' and 'despues' is when they effect the entire sentence (as adverbs I think). Por ejemplo:
Lo hice antes. = I did it before
Usually 'antes' and 'despues' will take prepositions after them (like 'de'). I think this happens when they are connecting two nouns like in Duo's sentence above (hands and dinner).
The dictionary meaning of lavarse is to wash, not to wash oneself or wash by me/you/one, so I think you would omit the reflexive pronoun in a literal (word-for-word) translation. You could just say
I wash the hands before the dinner. (literal)
Yes, it can be useful to do a word-for-word translation initially.
Supper means the meal late in the day. Dinner means the big meal of the day, which was formerly usually eaten around when we have lunch today. So traditionally the meanings were different.
Of course nowadays dinner is usually held late in the day. So sometimes the words can be interchangeable and sometimes not, depending on usage.