If someone quotes e.g. Leviticus 6:7, will this be the correct position no matter the language or translation?
In general, yes...
As Affable Geek answered, the divisions predate the printing press but were not part of the original texts. There aren't that many publishers of Bibles and most of them use the same system that English translations have used since Langton. So if you go to a Spanish-language service anywhere around the world, you can find your place in the text. (Assuming you understand Spanish, of course.)
It seems likely that this is for convenience and because of the network effect. Since most tools of interpretation (dictionaries, commentaries, concordances, etc.) use the standard system, most people who purchase a Bible will expect to find it to be compatible to that system. A publisher would quickly go out of business if they tried to introduce a novel system.
with two exceptions.
As H3br3wHamm3r81's answer notes that the Jewish Bible (what we call the Old Testament) uses a different system. Since Judaism has a separate apparatus of interpretation, Jewish publishers divide chapters and verses in slightly different places. It's not different enough to cause major problems, but we have experienced confusion over on Biblical Hermeneutics. One noticeable pattern is that Psalms in Jewish translations often make the short descriptive text (ascription) part or all of verse 1 in contrast to Christian translations. For this reason, I like to quote Bible passages or at least link to them rather than just include bare citations.
The earliest Greek translation of the Tanakh included a slightly different order of the Psalms then the Hebrew original. Since the standard system adheres to the Hebrew order, this is normally not a problem. But it's something to be aware of if you are talking Psalms across the Eastern/Western divide or if you are reading Catholic liturgy (which uses the Greek system for some reason).
As it turns out, Leviticus 6:7 is not the same between the Christian system and the Jewish system:
And the priest shall make atonement for him before the LORD, and he shall be forgiven for any of the things that one may do and thereby become guilty.”—Leviticus 6:7 (ESV)
And this is the law of the meal-offering: the sons of Aaron shall offer it before the LORD, in front of the altar.—Leviticus 6:7 (JPS)
It turns out the Jewish system puts what we call Leviticus 6:1-7 into the end of chapter 5 as verses 20 to 26.
Psalms 88:5 in King James Version:
Free among the dead, like the slain that lie in the grave, whom thou rememberest no more: and they are cut off from thy hand.
Psalms 88:5 in Jewish Publication Society (based on versification in Masoretic text):
I am counted with them that go down into the pit; I am become as a man that hath no help.
Even some verses differ in NT according to which Greek manuscript you are reading from.
Yes, generally, though not universally.
While definitely a respected theologian (he was an Archbishop!), his "sense"-based structure is by no means universal agreed upon. Indeed, there is an apocryphal tale underlying the seeming lack of any rhyme or reason in the structure that suggests Langton was riding his horse as he inserted the numbers. The story goes that when his horse's feet touched the ground, he inserted a verse, and when he fell off the horse, he inserted a chapter!
Other versification schemes exist, but his is the most common. Manuscripts that predate this convention are sometimes back-versed, and some translations have reversed the scripts, but most popular translations, such as the KJV, NIV, NASB, RSV, ESV, NLT, etc... all use Langton's schemes.
Indeed, the versification is so well received that as modern manuscripts cast doubt on the authenticity of some verses (see 1 John 5:7), many modern translations will omit the verse altogether, going directly from 6 to 8.
Internationally, those that choose to follow this pretty well received convention tend to follow it as well.