Hi there - lecturer in Latin literature, historian of 4-5th century AD, and have taught both Classical and early medieval history so have a fairly good background for this one. To take your questions one at a time:

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Latin becoming the only surviving Italic language:

In the end, this is about political and military power. Rome had totally dominated the Italian peninsula by the end of the third century BC, and so to be 'somebody' in Italian politics meant being tied into Roman political life and therefore speaking the Roman dialect, Latin. The same thing happens all the time, even in the modern world - why is 'standard English' the English of people living in England's southeast? Because that's where political power and wealth have historically been concentrated within the United Kingdom.

This being said, you do find occasional glimpses of the other Italic languages. The only one I can think of off the top of my head is that, during the Social War (91-87 BC), the Italians minted their coins in both Latin and Oscan. So we just have a few words here, but it shows that Oscan was still a language that had currency in the first century BC, but all our surviving texts come from the political and social elite of ROME, so they are Latin texts. You could also have a look at the Cippus Abellanus, an inscribed Oscan text from the 2nd century BC.

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How Vulgar Latin split off from Classical Latin:

Vulgar Latin never 'split' from Classical Latin. Vulgar Latin (from Latin 'vulgus', meaning common people) refers to a wide spectrum of 'everyday' Latins that would have actually been spoken by ordinary people. They are distinguished, in the classical period, from 'Classical Latin' in much the same way that we might today distinguish academic English from the way an ordinary person speaks, so they're fundamentally the same language, but working with different vocabularies and grammatical expectations. Again, you can easily see this relationship by modern analogy - my writing is always very 'proper' academic English, but if you took a transcription of me speaking it would sound very 'vulgar' - dropped h's, 'ungrammatical' constructions, non-standard words etc etc. And you can see this exact same thing - that Classical and vulgar were not separate branches, but part of the same language - occasionally even with Classical figures. Cicero is the gold standard for what good classical prose should look like. However, read some of his most intimate letters to close friends (especially Atticus), and you find a much more colloquial style - words get dropped, constructions simplified, and so on.

For examples of what vulgar Latin looked like, there's a number of good places to go. Graffiti are a really good one, and Pompeii and Herculaneum much the best place to view that - there's loads of publications that list graffiti found in these cities, and it will be much more representative of ordinary speech than are 'classical' texts. There's also the plays of Plautus (much much earlier but again capturing some of the rhythms of real speech) and the dinner scene in Petronius's 'Satyricon'. Another good source, from the fourth century AD, is the Vulgate Latin Bible, which again is written in a self-consciously colloquial style (and mirrors the Koine Greek of the original New Testament). And one could point to dozens more - the Vindolanda Tablets are another nice example.

To give a single example of what sort of changes we're talking about, I often use the Nicene Creed, which was written in 325 AD and has lots of nice 'vulgar elements'. Just to look at the very first clause, in the creed it says 'credo in unum deum' (I believe in one God). Now, this is technically not 'correct' - in strict Classical Latin this should be 'credo uno deo', but a more colloquial rule has been adopted in this form of the language. To be technical (if you know anything about grammatical structures) the creed uses a more 'modern' style of using a preposition with the verb (I believe IN one God), which is what the modern languages would do (including English), whereas the Classical version uses a dative case (which you can't really translate 'literally' into English - but credo uno deo/ credo in unum deum mean exactly the same thing).

So vulgar Latin is not 'a language' per se, but more an umbrella terms to encompass 'non-standard Latin' written during and immediately after the classical period.

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How Church Latin is different to Classical Latin:

Ok and then following on from the above, we get so-called 'Church Latin' as a product of vulgar Latin. Church Latin developed during the medieval period, particularly beginning with the Carolingians (9th century AD). What had been happening in the 500 years before this (and I'll talk about that more in a second) is that the spoken and written Latin of the western Mediterranean had been branching and changing, ultimately giving us the modern Romance languages. What the Carolingians did was to halt this process by imposing a highly artificial standard based on the Classics (Cicero and Vergil and their ilk). This meant that, from this point on, the spoken Romance languages and the written 'Church Latin' began to grow sharply apart and ultimately become mutually unintelligible. The Carolignians did this in order to ensure a fixed written standard across widely differing regions of the world, so a scholar writing at one end of Latin's spectrum of languages would still be intelligible to one writing at the other, and both would still be able to read the important works of classical antiquity. And from this new standard came church Latin.

Like vulgar Latin, Church Latin isn't one thing, but a whole range of different kinds of written Latin that use Classical standards but also impose different rules. It got a particular boost as a living, breathing language during the 12th century and the scholastic movement, when Church scholars began to free themselves more and more consciously from the limits of classical vocabulary and grammar and to develop their own rules for (particularly) word creation, allowing them to invent swathes of new words that enriched the (particularly) theological texts they were writing. It's hard to explain without getting very detailed, but someone trained on Classical Latin will find a high medieval text to be perfectly legible, but really odd, with loads of words that they can guess the meaning of (thanks largely to modern cognate versions) but which they will never have seen in a work written by a Roman. I always say that Church Latin is Latin written by people who are thinking in a modern language, but writing in the ancient one.

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How close today's Romance languages are to Vulgar Latin, including the minor languages such as Catalan and Sardinian:

The modern Romance languages are on a clear continuum with Classical Latin, and there's nothing shows this off better than the so called 'Oaths of Strasbourg', which is a record of the oaths sworn by a pair of Carolingian kings in 842. The oaths were sworn in both the Germanic and the Romance language spoken by the kings and their armies, and the Romance version of the oaths is often considered to be the first written example of Old French (as opposed to vulgar Latin). To see the evolutionary chain, I'll give you a line of those oaths written 1. in neat classical Latin, 2. as they appear in the original text, and 3. in modern French, and you can see that this 9th century vulgar Latin/Old French neatly straddles the two:

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1. pro dei amore et pro christiano populo et nostro communi salvamento
2. pro deo amur et pro chistian poblo et nostro commun salvament
3. pour l'amour de Dieu et pour le peuple chrétien et notre salut commun

(This means 'For the love of God and from the Christian people and our common salvation'). As you can see, there's a clear chain connection 1 to 3, and the bridge is our 9th century text.

Of the modern Romance languages, Spanish and Italian are very similar to Latin, French and Portuguese less so. I barely know Sardu at all, and don't know a single word of Catalan or Romanian, so I'm afraid I'm less clear on their relationships to Latin. My impression of Romanian has always been a very strongly Romance language, very closely related to Latin, but spoken with Slavic accentuation and stress patterns that make it sound rather different.

(I've run out of space so I'll do the last one in a separate comment)