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As in the title, should something like 不对 be written as bù duì (as in the individual characters) or as bú duì (the way it's actually spoken)?
I know that my teacher writes e.g. 你好 as nǐ hǎo (i.e. despite the ní hǎo pronunciation). Google Translate seems to stick to writing it according to the individual character tones too. Is this the standard way? I'm confused because I thought of Pinyin as 'just' a pronunciation guide, so why not write everything exactly as it is spoken?
I want to emphasize what @user3306356 says in a comment. Hanyu Pinyin (漢語拼音) is an officially established system and has explicit rules which you can see here. As @jdods remarks in a comment to this answer, that means tones are written as assigned and not as pronounced, while syllables are grouped as pronounced.
This is important because Hanyu Pinyin is not only a way to teach children in China to read, or a way to help foreign learners of Chinese. It is also an international standard for rendering Chinese texts in a western alphabet.
You should probably try to learn the actual international standard -- especially if you think you might later use Pinyin in transcribing anything for other people to read.
As to the question "why not write everything exactly as it is spoken?" Here are three answers. But in fact Hanyu Pinyin is the outcome of nearly a century of efforts by many scholars and linguists, and I am probably missing many of their reasons.
First, so far as two or three third tones in a row are concerned, Hanyu Pinyin does tell you exactly how they are normally spoken because you know the Hanyu Pinyin rules. What you really meant was "why not write each individual syllable exactly as spoken without relying on context?"
So, second, context is a very complex issue: As @michealyus notes, third tone sandhi on more than three successive third tones depends on how the sentence is meant to be broken into phrases. More: the tone on a word may turn neutral at the end of a phrase, or may not if you want to emphasize that word for some reason (indeed even so common a phrase as 你好 can be spoken as ni3 hao3 in a context that calls for for special emphasis). And the exact way a neutral tone is pronounced depends on the tone before it (see this link). Thus, many words would have at least three spellings. Now suppose you as writer want to quote something someone said. You would have to choose how to spell each sentence by deciding how the sentence would have been broken into phrases (this is often ambiguous) and how much emphasis was intended (ditto).
Third, from the reader's viewpoint, consider the writer whose pseudonym is spoken lao2 she3. Is the first word there a second tone word? Or is it a third tone word being written as second? Better to write it as Lao3 She3, and use the rule for third tone sandhi. The ambiguities of phonetic representation of Mandarin are bad enough intrinsically. Let's not make them worse by giving single words multiple spellings.
It is a common convention writing pinyin with the dictionary tone. Students will soon learn the rules for tone sandhi, and will then apply it to both pinyin and characters. If applying tone sandhi in pinyin, there is a risk of associating a certain syllable with the wrong meaning or learning the wrong tone.
So yes, nǐ hǎo is the standard rendering in pinyin, despite the actual pronunciation ní hǎo.
If it makes up the same "expression" (usually ranging from one to four words) in pinyin, such as "budui', "nihao", etc, it is put together.
If it is not the same "expression", they will be separated. For example: hen gaoxing renshi ni. Notice "hen" is separate from "gaoxing" and so forth.
This helps the reader read with ease rather than having to make sense of a bunch of random letters, which, for a beginner, they would not be used to yet.
As you already know it, there are five tones of Chinese character pronunciations, the level tone(阴平), the rising tone(阳平), the falling-rising tone(上声), the falling tone(去声), and the light tone(轻声).
For the problem of "你好", it's the problem that it's hard to pronounce a sequence of multiple falling-rising tones continuously, if you are not pronouncing it slow and carefully. All characters before the last one will be likely to pronounced as rising tones, and the last character, if you pronounce it shortly, it will likely to pronounced as a falling tone, and you can extend the length and complete a falling-rising tone. The reason is the falling-rising tone has the weight just a little greater than the light tone, the real pronunciation will effected by the tone before and after it. If you pronounce "你好" slow and clearly, you can pronounce both character as the correct falling-rising tone.
"不" is essentially pronounced as falling tone, but when using before another falling tone character, it pronounces as a rising tone, which is because of the classical Chinese feature of "rhyming", witch is almost lost at all on the character pronunciations in modern Chinese, but still exists in poems --the construct of characters, and a little of very frequently used characters.
There are also polyphonic characters in Chinese, which has multiple pronunciations with difference meanings. Witch is similar to English words that change their meaning depending on stress position. The different pronounces of a polyphonic character can't replace each other, most of them has different meanings on each pronunciations, there are often some relations between these meanings, but can't be replaced each other.
