Have no fear, you Resident Dutchie is here 
You are touching one of the finer points of our language here. Let’s use your example: Elizabeth ten Grotenhuis.
Alphabetise it under G. Sometimes it is seen as “Ten Grotenhuis, Elizabeth”. Although most of the times in libraries, etc.: “Grotenhuis, Elizabeth ten”.
When using a name in sentences:
“Elizabeth ten Grotenhuis says…”
“Ten Grotenhuis says in her last book…”
“In her last book, Ten Grotenhuis says…”
So the capital T is only dropped when the name is written out in full. HOWEVER this is only the case for Dutch names. Flemish names (i.e. names from the Dutch-speaking part of Belgium) have different rules. Here, the “Van”, “Ter”, “Ten”, etc. is ALWAYS capitalised, even if the name is written in full. Why, I do not know, but the Belgians are funny that way 
Van Hemmessen and De Vries would follow the very same rules in alphabetisation (?) and capitalisation.
In Dutch, the first word is ALWAYS capitalised, like in your language. But as seen, when dropping the first name, capitalisation already takes place.
Polycarp, here’s something few people know: the full name is Ludwig van Beethoven. Yes, “van” with an “a”. His parents were Dutch, I believe. Could have been his grandparents too. And officially, that name should be treated the same as above. But since “Beethoven” is so widely used, not even the Dutch bother to do it… if they KNOW, that is 
In Poly’s example, “Van Gogh” is correctly uppercased, but only because the first name isn’t there. Hence, it would be “Vincent van Gogh”. Popularity is not an issue here 
de Maupassant is written that way because the French rules differ, I think - I’m not even sure if they do. Either way, it’s not because of the most widespread use of the name.
As for the meanings - it’s Old Dutch mostly, but I’ll try and convert it to modern English:
“van” would be “of”. So, Jan van Amsterdam is Jan of Amsterdam (or John, if you will).
“ter” and “ten” can be best translated as “on the”, “to the” or “near the”. Hence, Piet ter Duin is “Pete near the Dunes”. Although it doesn;t really mean that in Modern Dutch though.
“van de” and “van der” are variations of “of the”. Thus, Henk van der Molen is “Henk of the Mill”. “from the” is sometimes a better translation.
I’m not even going into double last names and nobility names… that’s gonna take me all week

I hope this clarifies somewhat.
Coldfire
“You know how complex women are”