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I am pretty sure that when native speakers talk about _speed_ , they say "速さ",
"広さ" for _width_ , "大きさ" for _size_ , etc. Am I correct about this? If I am,
then:

Why not say "遅さ" instead of "速さ", "狭さ" instead of "広さ"? I suspect that when
you convert comparable adjectives to nouns, you should use the greater of the
two. However, surely this must just be a convention, right? So,  
"新幹線の速さは何ですか？" // wrong?  
"新幹線の遅さは何ですか？" // wrong?

what about:  
"snailの速さは何ですか？" // wrong?  
"snailの遅さは何ですか？" // wrong?

And finally, are there cases, if there are any, where it's natural to use
"遅さ", "狭さ", "小ささ", etc.?

thanks.
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word-choice adjectives
why "速さ" instead of "遅さ"、etc
Unless I've missed something, it is because it sounds as odd in Japanese as it
does in English. "新幹線の速さは何ですか？" - what is the speed of the bullet train?
"新幹線の遅さは何ですか？" - what is the slowness of the bullet train?

Similarly, 広い is more like 'spacious' or 'extent'. 狭い is 'narrow'. If you
attempt the higher JLPT levels (well, 3 & above) choosing the specific word is
vital.
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As you say, it is a convention to use the more "positive" word in Japanese.
However, there are situations where we native speakers opt to use the "other"
word. I could think of two conditions that make using the other word sound
natural or even better.

1 Common knowledge:

This is when a thing's slowness, smallness, lightness, etc. is commonly known
to the general public and the speaker decides that it can be used as a premise
of a conversation. The slow speed of a snail （遅さ） is a prime example of this.

2 Contextual/Situational:

If you have to talk about the slow speed of a snail to someone who has never
seen or heard of a snail, you would need to first explain how slow it moves.
After explaining it, you can logically use 遅さ instead of 速さ.

Another example would be how "slowly" a Shinkansen train which is well-known
for its high speed could run on a snowy day. It can actually run more slowly
than a bicycle in certain places on a snowy day. In this case, you first
explain the weather condition and you can use 遅さ with no problems or
unnaturalness to the native ears.

Finally, allow me to correct your Japanese. You cannot say 新幹線の速さは何ですか as that
clearly is a direct translation from English. You would say
新幹線の速さはどれぐらいですか、新幹線のスピードは何キロくらいですか, etc. Since we are discussing 速さ/遅さ, I
probably should not be using the word スピード even though it is the more natural
word choice for us.
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Be careful regarding nouns vs. adjectives.

In your case, `速さ` = `speed` is a noun, while `速い` = `fast` and `遅い` = `slow`
are adjectives.

  1. This is basically same as in English.

In English, you do not say `What is fast of 新幹線?` or `What is slow of 新幹線?`.
Instead, you say `What is the speed of 新幹線?` = `新幹線の速さはなんですか?`

This is correct sentence, but not perfect. (It sounds weird.)

  2. The speed of a 新幹線 is not stable (while, e.g., the maximum speed of a 新幹線 is stable), so it is more common to ask about the speed using an adjective: `How fast is 新幹線?`, `How slow is 新幹線?`

But 新幹線 is obviously a fast vehicle, and you want to ask about fastness. So
the best sentence is `How fast is 新幹線?` = `新幹線はどれくらい速いですか?`

You can apply this same strategy to the snail sentence:

`snailの速さは何ですか?` is not wrong, but not perfect. You want to ask slowness of a
snail because it is a slow animal, so the best sentence is

> "How slow is a snail?"  
>  snailはどれくらい遅いですか?
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