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[eHalcyon]

I know he doesn't like Mine, but I can't quite figure out his opinion on Mint.

Exactly.  Also not sure which he prefers between cyan and gray.

[eHalcyon]

He likes Sony but not Toshiba.

Actually, I think this one has to be wrong too, unless your cousin likes pens and whips.

A pen is not a penny.

Presumably he likes gray about as much as he likes Sony?

If he likes Sony then he probably shouldn't like yellow, except Kuildeous says he does.

[chwhite]

He likes Sony but not Toshiba.

Actually, I think this one has to be wrong too, unless your cousin likes pens and whips.

Plus the spoiler with yellow vs. gray... the original has "flypaper" which should be just as certain as yellow, no?

I would expect him to like gray but NOT flypaper.  However I would also expect him to like Yale but not Princeton.  At least if usage matters.

[cayvie]

He likes Sony but not Toshiba.

Actually, I think this one has to be wrong too, unless your cousin likes pens and whips.

A pen is not a penny.

Presumably he likes gray about as much as he likes Sony?

If he likes Sony then he probably shouldn't like yellow, except Kuildeous says he does.

the y in Sony is a vowel; the y in Yellow is an approximant

[Grujah]

He likes Sony but not Toshiba.

Actually, I think this one has to be wrong too, unless your cousin likes pens and whips.

A pen is not a penny.

Presumably he likes gray about as much as he likes Sony?

If he likes Sony then he probably shouldn't like yellow, except Kuildeous says he does.

the y in Sony is a vowel; the y in Yellow is an approximant

I know it has soemthing to do with vowels and consonants but I don't know what 

[Morgrim7]

He likes Sony but not Toshiba.

Actually, I think this one has to be wrong too, unless your cousin likes pens and whips.

A pen is not a penny.

Presumably he likes gray about as much as he likes Sony?

If he likes Sony then he probably shouldn't like yellow, except Kuildeous says he does.

the y in Sony is a vowel; the y in Yellow is an approximant

I know it has soemthing to do with vowels and consonants but I don't know what 

Me neither!

[blueblimp]

He wants answers, not questions.

[Morgrim7]

He wants answers, not questions.

That didn't help.

[cayvie]

he wants buffy, willow AND xander

[Kirian]

Non-native speakers/writers:  total spoiler.  Everyone else has been warned.

In English, five letter are considered proper vowels--A, E, I, O, and U.  The letter Y can sometimes be considered a vowel (often at the end of a word or within a diphthong, i.e. the Y in "mayhem") but most of the time is a "consonant," though in linguistics terms technically an approximant (almost all initial Ys are approximants, including when they act to palatalize the next vowel, as in "yellow" for instance.  The letter W is also an approximant, but can only be found as a vowel when part of a diphthong, and thus is never a vowel on its own.

In the Latin alphabet, A, E, I, O, U, and Y are in that order in the alphabet.

Each word the cousin likes (1) has more than one vowel, (2) has no repeats, and (3) has its vowels all in alphabetical order.

However, there is a distinction (as in many other languages) between vowel letters and vowel sounds.  In English, the latter would include digraphs like "ai" and "ay", both of which usually rhyme with the a in "rate."  The word "gray" is semi-ambiguous:  is the letter "y" in gray a separate vowel or not for the purposes of the puzzle?  Or, more specifically, is the letter Y considered a vowel when it is part of a digraph?  If not a separate vowel, then condition (1) is false and the word fails.

This is resolved by the word "chain" in the OP.  Despite the word having only one vowel sound (the digraph "ai"), it has two vowel letters.  The "y" in "gray" is pronounced as part of the digraph "ay," and thus should have the same status as the "i" in "chain."  In particular, "gra," were it a word, would have a different vowel sound (see "bra"); the Y changes the mature of the vowel.  Similarly, the "e" in "apple" is silent, but still counts as a vowel letter.  Meanwhile, the word "yellow" counts because the Y is pronounced as an approximant, but flypaper does not because the Y is pronounced as a vowel.

In order to be properly unambiguous, then, the rule as I state it above must technically include the phrase "where 'vowel' is defined as a single letter pronounced as a vowel, including one that is part of a digraph.

See also: http://en.wikipedia.org/wiki/English_orthography

Whew! 

[Kirian]

he wants buffy, willow AND xander

Hey, who doesn't?

[Young Nick]

How did I not get this? I feel dumb. Sheesh.

[Grujah]

Had to stop reading at one point.
This is so nonsensical.
My language has 1 letter == 1 sound rule. End of Story.

[Kirian]

Had to stop reading at one point.
This is so nonsensical.
My language has 1 letter == 1 sound rule. End of Story.

Of interest, which language?  The only ones I know of off the top of my head are Spanish and Italian.

[Morgrim7]

How did I not get this? I feel dumb. Sheesh.

Me too.
King's Court but not Throne Room
Alchemist but not Laboratory
Hamlet but not Farming Village
Bishop but not Monument
Market but not Grand Market
Forum but not Isotropic

[eHalcyon]

the y in Sony is a vowel; the y in Yellow is an approximant[/spoiler]

Whoooops.  I feel dumb now.

[Grujah]

Had to stop reading at one point.
This is so nonsensical.
My language has 1 letter == 1 sound rule. End of Story.

Of interest, which language?  The only ones I know of off the top of my head are Spanish and Italian.

Serbian.

I don't think Spanish and Italian do this. Better than English still

[Kirian]

Had to stop reading at one point.
This is so nonsensical.
My language has 1 letter == 1 sound rule. End of Story.

Of interest, which language?  The only ones I know of off the top of my head are Spanish and Italian.

Serbian.

I don't think Spanish and Italian do this. Better than English still

Looks like Spanish is better than Serbian at it, actually, but not Italian.  (Using Wiki here, though, so you can probably tell me if Serbian actually uses vowel tones or if it's a "high grammar" thing.)  English and French are the absolute worst though.  English has twelve vowel sounds and at least ten diphthongs, all represented by combinations of seven letters!

[Grujah]

Vowel tones? What is that? I hear it for the first time.
Like soft and hard in russian? We don't have that.

I think not. Spanish I think still has "ch" (pronounced as in english). We don't. We use č (ч in Cyrillic). They also have stuff like ll in Castilla. For that we use љ in Cyrillic alphabet and because Latin alphabet sucks we need to use two symbols, lj (but this is still a single letter, like, it is in alphabet as "lj" and when you do crosswords you put "lj" in one field. One letter just 2 symbols for it. You never have j after l in Serbian anyway).

[Grujah]

Like, if you go here:
http://en.wikipedia.org/wiki/Phonemic_orthography#Comparison_between_languages

There is a list of phonemic orthography languages, and out of all on the list only we are the few that dont have" (apart from this, that, and that)" clause Bulgarian doesn't have it either.

[Kirian]

Vowel tones? What is that? I hear it for the first time.
Like soft and hard in russian? We don't have that.

Amusingly, from your link:  "Serbian and Croatian do not distinguish tone and vowel length"

But I was mainly looking at vowels rather than consonants.  South Slavic languages apparently do have almost perfect consonant orthography, partially (as you note) thanks to the Cyrillic alphabet.

[cayvie]

There are many Englishes and they have all sorts of different vowel sounds

[Grujah]

Vowel tones? What is that? I hear it for the first time.
Like soft and hard in russian? We don't have that.

Amusingly, from your link:  "Serbian and Croatian do not distinguish tone and vowel length"

Yeah, now that I think about what it is, no we don't. We don't have soft and hard "e" and similar stuff, so we are not used to it and sound like Neanderthals when speaking English. 

[Kuildeous]

Kirian's exquisitely detailed solution is correct.

The reason I accidentally muddied the waters is because there are differing takes on whether Y is a vowel in words like "boy" and "day." I found inconsistent results on Google and realized that this is a point of contention. It's obviously a vowel in "city" and obviously a consonant in "vineyard." I had thought my spoiler might give something away because it draws attention to Y. That's not necessarily true, I see.

If nobody mentioned it, I was going to say that my cousin liked facetiously but not jokingly. In my mind, I thought that would have been a big giveaway but maybe not.

[Kirian]

Kirian's exquisitely detailed solution is correct.

The reason I accidentally muddied the waters is because there are differing takes on whether Y is a vowel in words like "boy" and "day." I found inconsistent results on Google and realized that this is a point of contention. It's obviously a vowel in "city" and obviously a consonant in "vineyard." I had thought my spoiler might give something away because it draws attention to Y. That's not necessarily true, I see.

If nobody mentioned it, I was going to say that my cousin liked facetiously but not jokingly. In my mind, I thought that would have been a big giveaway but maybe not.

Yeah, there aren't too many people who know that "facetiously" and "abstemiously" are the two words in the English language with all six vowels in alphabetical order, so not that big a giveaway.