… I don’t get it.
Okay, so if Alice’s number is 6, then Bob’s could be 12, or 3. But if Alice’s number is 8, then Bob’s could be 16 or 4. Or 10, 20, and 5…
Here’s how I work it out. No guarantees of correctness.
⓪ A is in [1…30]; B is in [1…30]
Alice: “Is your number double mine?”
Bob: “I don’t know."
① B is in [2, 4, …, 30]
Bob: “Is your number double mine?”
Alice: “I don’t know.”
② Giiven that Alice knows ①, A is in [4, 8, …, 28]
Alice: “Is your number half mine?”
Bob: “I don’t know.”
③ B is in [2, 4, …, 14] (must be even following ①)
Bob: "Is your number half mine?”
Alice: “I don’t know.”
④ A is in [1…7]
but looking at ② above A is also in [4, 8, …] and the only intersection is A=4.
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Here is my best hint (especially for anyone who is unsure how to start with this) without giving the full explanation:
The answer “I don’t know” can be rephrased as "I cannot say with certainty that the answer is “yes” and I cannot say with certainty that the answer is “no”. One of those two things carries a lot more information than the other.
I loved this partly because at face value the conversation sounds so useless : )
My second hint:
The information accumulates. At each point Alice and Bob can each assume that the current question and answer takes the previous answers into account. Tracking everything that each of them knows at each step on paper is a good idea. (Edit: @COMaestro has detailed a pitfall which occurs when you assume that new information is being conveyed by the questions themselves; this assumption can unfortunately break the puzzle, so you should consider only the answers as providing new information.)
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My answer: Is it 8?
Working
For any number to possibly be double the other, it has to be even.
For any number to possibly be half the other, it has to be below 15.
For both numbers to be even and to be possibly a double of the other, half the number would also have to halve into an even number.
eg 12, 10, 6, and 2 all split into odd numbers. 4 halves to 2, which would halve to 1 and therefore doesn’t fit. 8 halves to 4, which halves to 2.
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I can tell you that @RogerBW has the correct answer (which is different). You’ve taken a slightly less-rigorous approach and managed to trip yourself up.
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Okay, I am going to take a crack at this, and assume perfect logic on the questioners’ parts.
So Alice asks Bob if his number is double hers, which suggests her number is 15 or less, as anything larger than that would be outside the range. Bob doesn’t know, which suggests his number must be even, as an odd number could not be the double of another number, and he would have said “No.”
Bob asks if Alice’s number is double his. Since Bob knows her number is 15 or less, he would not ask this question unless his number was 2, 4, or 6, as it is even and anything higher would result in a number over 15. Since she doesn’t know, we know her number is also even, and must be 4, 8, or 12.
Alice then asks if Bob’s number is half of hers and he doesn’t know, as she could be 4, 8, or 12. Bob asks the same of her and she also doesn’t know, which is a paradox, as none of her possible numbers are half of Bob’s possible numbers, so she should have said “No.”
So I am left disappointed with Alice and Bob.
Instead I will do the other method.
If Bob doesn’t know if his number is double Alice’s, that means his number is one of the even numbers between 1 and 30. So when he asks Alice if her number is double his and she doesn’t know, it means her numbers go by 4’s (so 4, 8, 12, etc).
She asks if Bob’s number is half of hers and he doesn’t know, so his numbers reduce to the even numbers between 2 and 14. He asks if Alice’s number is half of his and she doesn’t know, that just leaves the 4, as out of all his potential numbers, 4 is the only one that can be half of one number in Bob’s set.
So the answer is 4, but poorly determined. In my opinion. 
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I had a similar situation in my first working out where, if operating along a linear timeline using perfect logic, Bob knows that Alice’s number cannot be half of his.
And then I gave up and read the spoiler responses.
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Now you tell me. 
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Sorry about that – I wasn’t smart enough to break the puzzle that way! :)
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I keep doing them. Short one this year, though.
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Anyone had a stab at this yet? Nice little distraction for a lunch break.
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This was fun! I did not figure out all of the puzzles, but got enough to get the overall solution.
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I haven’t even taken a stab. I do like these things but a quick survey made it look like Mensa style riddles. Which I always struggle with, because the questions and answers are often both ambiguous and you don’t know when you have the right answer, or you have one that seems right and they say “no, sorry, we wanted to take that in a completely different direction.”
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I’ve only had a quick look myself, but it looks like they vary a bit. And it’s aimed at least partly at schools, so shouldn’t be excessively unsolvable!
I found them an interesting blend of “work out what the solution process must be” and “actually apply the solution process”. So far I’ve done 1, 4 (my favourite), half of 5, and 7.
One thing that helped with this one is every answer is something that could follow the word, “Christmas”, so you have a theme to work with and if you get something outside of that, you know you did something wrong.
I definitively solved 1, 2, 4 and 6. I figured out about 2/3 of 3, which was enough for me to get the answer, and have no clue on 5 or 7, other than I would likely need to figure out the cipher for 7, and I just don’t have the time or patience for it.
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fourTH, thiRD, secoND, firST
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ETELPMOC (complete backwards, backed into this using the other three)
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WvU,t,SrQ,p,OnM,l,KjI (reverse alpha, every second)
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wrappiNG
Total word STOCKING
OK, that was pretty good. I’ll have to stab the others too.
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Okay, I shouldn’t have said “No clue on 5,” as I did get the third and fourth ones, it was the first two I had no idea on. 
I am in fact enjoying these. The one that initially turned me off is still a complete stumper - # 3. I always assume there is some background info I might need, or some arbitrary solution. I’m looking at it and it’s like, Jasmine. Herb/flower. Scarborough - parsley sage rosemary and thyme? Is that it? A Penny is a cent. Are those three words joined by “scent”? Did they go that way?
I have zero context on Declan or Pudsey, if they mean anything at all. So trailing y? Counting letters? Linking vowels?
I just have no idea how to slice that one.
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There might be a touch of background for this one, and likely works better if you are in the UK or know it well.
So, here’s one hint. Sirius and Pitch are two of three words that go together. The third word I didn’t know, but figured the answer for them out from just these two. After seeing the solution, I figure you need to be British or collect stamps to know the third word that belongs in this group.
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