Russian doesn't have a direct equivalent to the sound of English h.
But to some English speakers it seems odd that words taken into Russian from English which have an h sound are transliterated using the letter г, which sounds like English g rather than х, which while also not perfect, sounds a lot more like h.
Harry Potter → Гарри Поттер (garri potter)
So my question is, are there also words where х is in fact used to transliterate h, or is г always used?
The transliteration rules changed with time in Russian.
For instance, Daniel Defoe is Даниэль Дэфо (but Daniel Craig is Дэниэл Крейг); Thackeray, Disney and Halley are Теккерей, Дисней and Галлей (but P. J. Harvey is Полли Харви) and so on.
The names with well-established rules like Harry and Henry are almost always transliterated as Гарри and Генри; less common names like Harold and Harris can be transliterared both ways (Гарольд и Хэролд, Гаррис and Хэррис), modern last names (like Harvey above) are almost always transliterated phonetically.
Answering your question: modern transliteration rules (as defined by the classic work Старостин, Гиляревский. Иностранные имена и названия в русском тексте) prescribe to transliterate English fricative h as Cyrillic х, unless there is an established tradition to transliterate the name otherwise.
This transliteration is a rule, not an exception.
The book does not provide a complete list of traditional names, but includes Гамильтон (Hamilton), Гайд-Парк (Hyde Park), Герберт (Herbert) and Говард (Howard).
I was asked a similar question by Germans, who are surprised by our transliteration of Hans and Herz as Ганс and Герц.
To answer it directly, yes. For example, Helene is often written as Хелен (Хелен Келлер, Хелен Хант). The Hampshire county name is spelled either with х or with г. The words high tech, hi-end and HiFi are never pronounced otherwise than хай-тек, хай-энд, and хай-фай (or хифи).
In general, as far as I can tell, it depends on when a particular word came to the Russian language (as already mentioned by Quassnoi) and if there are any established rules. An interesting example is the German Heil Hitler which is spelled as Хайль Гитлер (so a mixture of the two styles). The reason is that the word heil was not known in Russian until WWII, and therefore it's just spelled as it's heard in the twentieth century. The name Hitler on the other hand follows the traditional German-Russian spelling, which dates back to the time of Lomonosov and probably stems from the dialects spoken by the first German immigrants in Russia and, at the same time, the old pronunciation of г in Russian (still often mixed with х by less educated people).
Some additional examples that come to mind --
Not a g/h issue, but just to show an example of transliteration variants. A friend of Sherlock Holmes - Dr.Watson may be spelt as Ватсон or Уотсон, the later one often seen in old (Soviet) translations of Arthur Conan Doyle's books.
Sir Winston Churchill is always translated as сэр Уинстон Черчилль, but Winston cigarettes brand always translated as Винстон.
P.S. In the German language ei is always pronounced as ай. The names of the towns Heidelberg and Mannheim will be correctly pronounced as Хайдельберг и Маннхайм not as Гейдельберг и Мангейм, as we have seen these on Russian maps.
Russian "х" sounds kind of like the Latin "H" except with a more guttural sound (like Hebrew "ח" except less harsh). Your problem with "г" is that in Ukrainian, "г" is equivalent to Latin "H" , and Russian "г" is equivalent to "ґ". That's probably a bit confusing, but I hope it helps.
