They all mean the same, yes. But use 'as' and 'since' for reasons your listener already knows only. Do not use them to introduce a previously unknown reason. The listener here can see that it is getting dark so 'as' is appropriate.

In your example the word "as" is being used to describe doing something (going home) in relation to (as) it getting dark. Other ways to word it that show what I mean - While it's getting dark we should go home During the process of it getting dark we should go home "Since" and "because" are synonyms of one another. If you replace "as" with "since" or "because" in your example (Since/because it's getting dark we should go home) it changes the meaning slightly. Other ways to write it would be - Due to the fact it's getting dark we should go home As a result of the fact it's getting dark we should go home

No difference. All three would have the same meaning. However, in a case such as this, it's more common to use 'as' or 'since' rather than 'because'. Personally, I would use 'since'.