Yes, it's where she lived, but it could have doubled as a family name.
Right, but as shown in the above link to the royal website explaining royals and surnames, as a royalty, Alix wouldn't have needed to have a surname - sometimes royalties use surnames (e.g. the Windsors) but more often than not, they are simply identified by the country/region they are from. In Alix's case, it was "von Hesse und bei Rhine." For many royalties it is a problem for even them to find out what sort of "surname" they can take if they previously haven't used one and suddenly need one. E.g. when Prince Philip had to give up his greek title when marrying into the British royal family -- no one knew what sort of last name he actually had. Was it "Schleswig-Holstein-Sonderburg-Glucksberg....?" Perhaps, but he was known as "of Greece" more commonly. So he ended up adopting "Mountbatten."
I have read that Queen Victoria and Prince Albert spoke German with the family which is why her daughters had no trouble marrying into German houses.
According to Victoria herself, her own native language was English, as she was not allowed to speak German when she was little, and became fluent later. When raising her own children, she insisted that she and her husband "spoke English just as much as German" amongst their children and family. Considering her husband was at first unsure of himself in English, and that German was his native tongue, it is natural that the children would have grown up bilingual. I think this would have been the same with Alix -- of course she was born and raised in Germany, and her English mother died when she was only 6, but still, given her closeness to her grandmother and her preference for English, I think we can say that perhaps both German and English were her "native" languages, with English being the preferred one. (I live in Canada and know a few people who don't actually know if their "native" tongue is French or English - they are just bilingual and don't have an accent in either language).
She could have learned at lot in the 22 years she was on the throne, but at the end she was still speaking English to her husband and her children.
I don't really believe that speaking English to her husband and children was a deliberate snub at refusing to speak the Russian language by Alix. My own grandparents immigrated to Canada from Germany 60 years ago, and although by now they speak English most of the time and have mastered the language, between the two of them and sometimes to their children, they
still speak German. The thing is, with the people to whom you are closest, you're going to want to express yourself the best you can - so it's logical and understandable you would speak to them in the language you feel the most at home with - not one that you only began learning as an adult (unless of course you have no other choice and no one understands your native tongue).