I've seen Genghis Khan spelled a number of different ways (Genghis and Chingiz are the two I've encountered most often) in various sources, and as I understand it, there's no one standard way to translate Mongolian names to English. Is one or the other of these spellings out-dated? Does the English-language historical community have a preferred spelling for this name?
I can't quite answer your question about current consensus among English-language historians, but I am qualified to discuss spelling and transliteration of Mongolian.
The spelling "Genghis Khan" has an unclear origin. Weatherford, for all he's worth, suggests it comes from a transliteration found in Persian sources.
The name in Modern Mongolian is Чингис Хаан, which I would transliterate as Chingis Khaan, though some variation is reasonable. Let me explain a few of the Cyrillic letters:
Ч is definitely a 'ch' as in the start of 'church' sound.
Sometimes you see versions where the 'g' is double: Chinggis. This is really about the syllable structure.
Х is probably the trickiest for foreigners. In English it's more guttural than 'h', but less hard than 'k'.
а is doubled indicating a long vowel, but not held excessively long. So I can definitely see an argument for Khan instead of Khaan which might lead to people over-emphasising the vowel length.
The transliteration Genghis is very well established, which is why it's so hard to displace, but it is considered outdated/unreflective of likely historical pronunciation of the name.
Edit
I should add that the final 'n' (Mongolian н) is pronounced as a velar nasal when in a final position, so like /ŋ/ or 'ng'. But transliterating as 'ng' often leads English speakers to make more of the 'g' than is really there. I forgot about this aspect because I am used to it! If you say "Chingis khaan" with just a straight 'n' sound it's more like хаана which is the question word for "where?"
You are correct that there is no shared alphabet between the languages Mongolian and English. Furthermore, there is a difference between ancient and modern Mongolian. However, it seems the spelling "Chingis" is more faithful to the Mongolian pronunciation. The spelling "Genghis" seems to be an artifact of it being derived from a representation of Chingis Khan's name in Arabic, which lacks a "ch" sound.
Chingis/Chinghis are really probably the closest in English to it. For what it's worth, the Mandarin version is 成吉思汗 Cheng-Ji-Si-Han which is basically just an approximation of the Mongolian pronunciation.