A community wiki to catalogue progress on this puzzle so far in the PSE Advent Calendar 2024 chatroom...
From the OP's hint, it looks like this string of emojis can be separated into four mini-puzzles to produce a word each, and one final meta-puzzle to make use of them. All mini-puzzle answers have now been found - just the meta to go:
1. MAST
2. FOUR
3. COMB
4. BOUND
Mini-puzzle 1 (solved in the chatroom by @Stiv):
This section conceals...
...a word ladder puzzle, where a series of 4-letter words are presented, each differing from the previous one by the changing of a single letter.
The preamble "Island greeting to Santa's scroll" implies that we are aiming to change the letters of MELE (as in the Hawaiian-themed Christmas song, Mele Kalikimaka) to LIST. The other emojis work out as:
MELE - MALE - SALE - SALT - HALT - HART - MART - ???? - MIST - LIST
Our answer must therefore fill the question marks, giving us either the nonsensical MIRT or the more likely option of MAST!
Mini-puzzle 2 (solved in the chatroom by @user39583 and @Bladeski):
This section needs to be interpreted by...
...looking up the Unicode numbers for these special characters (usually of the form "U+####" where each hash symbol is a digit 0-9), splitting the four digits into two pairs of two, and then decoding those number pairs using the A1Z26 substitution (01=A, 02=B, ..., 26=Z).
The symbols therefore decode as follows (presented as an image rather than text since some of these characters are not supported in all browsers):
The letters produced then spell out 'WISEMENREDNOSEREINDEER', or - with spacing and punctuation added - WISE MEN, RED-NOSE REINDEER.
What do we do with this? Look at the three symbols immediately preceding this run of emojis:
"+" indicates addition, "#" indicates a number, and ":" is a separator showing that this should apply to what follows. This suggests we should add the number of wise men to the number of red-nosed reindeer.
The answer to this is FOUR (3 wise men - traditionally - in the New Testament story of the magi, and only 1 Rudolph!).
Mini-puzzle 3 (solved in the chatroom by @Bladeski and @Stiv):
This section conceals a...
nonogram puzzle. Many of these emojis can be linked to the seasonal song The Twelve Days of Christmas, and particular days:
Day 1: A partridge in a pear tree (the tree emoji)
Day 2: Two turtle doves (the turtle emoji)
Day 3: Three French hens (the chicken emoji)
Day 4: Four calling birds (the telephone emoji)
Day 5: Five gold rings (the ring emoji)
Day 6: Six geese a-laying (the goose emoji)
If we substitute each of these emojis with the number of the day on which they were first gifted, and treat the gift emojis as just dividers, then the downwards and rightwards arrows suggest that what follows after them can be interpreted as the down and across clues for a nonogram puzzle. The resulting nonogram is uniquely solvable:
The final two emojis in this section - ββ - instruct us to divide the resulting grid into 4 equal squares and interpret each as if the white cells indicate the positions of a signaller's arms in flag semaphore. Doing so gives us the four letter answer COMB.
Mini-puzzle 4 (solved in the chatroom by @user39583 and @Bladeski):
This section conceals a puzzle involving...
...extracting indexed letters from words clued by the emojis. The question provides an example first (preceding the double colon, as in an analogy problem) and then provides another poser for us to solve.
The example tells us that "naughty gift" - meaning COAL - is clued by the 3rd letter of a word clued by the playing card emoji, the 2nd letter of a word clued by the number '2', the 2nd letter of a word clued by the rainbow emoji, and the 4th letter of a word clued by the troll emoji.
To find what these emojis represent, we will need to consider...
...the lyrics of the traditional Christmas carol 'Deck the Halls'. The first verse (in one popular variant) goes:
Deck the hall with boughs of holly,
'Tis the season to be jolly,
Don we now our gay apparel,
Troll the ancient Christmas carol.
(I have omitted all the fa-la-la-la-la-ing for brevity!)
Note the words emphasised in bold: DECK, TO, GAY, and TROLL. These are the words suggested by the emojis mentioned above! And if we index into them as suggested we get: DE{C}K, T{O}, G{A}Y, and TRO{L}L, or C-O-A-L as required! Bingo!
Now consider further lyrics from the carol...
See the blazing yule before us,
Strike the harp and join the chorus.
Follow me in merry measure,
While I tell of Christmas treasure.
Fast away the old year passes,
Hail the new, ye lads and lasses!
Sing we joyous all together,
Heedless of the wind and weather.
In every line of verses 2 and 3 there is a word (in bold here) that corresponds to one of the emojis, some paired with a number (emphasising the letter index to be extracted from the word) and some with an 'X' (implying no letter is to be taken from these). This gives us:
1st from flame emoji = {B}LAZING
3rd from musical note emoji = CH{O}RUS
5th from ruler emoji = MEAS{U}RE
none from coin emoji = TREASURE
none from old man emoji = OLD
1st from Greek letter nu = {N}EW
none from mathematical 'all' symbol = ALL
4th from wind elemental symbol = WIN{D}
The 5 letters extracted give us the answer 'BOUND'!
Final metapuzzle:
The use of 'KEYS' here is doing double duty - not only is this section providing us with the keys to the metapuzzle, it's also alluding towards the keys of a piano or other musical instrument, because these sets of numbers represent notes on a keyboard (or similar) and sound out lines from particular Christmas songs and carols...
The first set of 'key' numbers 3455165421 corresponds to the opening notes of "It's the MOST wonderful time of the year" (when 1=C, 2=D, 3=E, etc.). This corresponds to the first puzzle's answer 'MAST' with an O/A misprint.
In similar fashion, the second set - 12333651 - is a line from White Christmas: "May YOUR days be merry and bright". This corresponds to the second puzzle's answer 'FOUR' with a Y/F misprint.
The third set - 543432312765 - is a line from O Come All Ye Faithful: "COME and behold him, born the king of angels". This corresponds to the third puzzle's answer 'COMB' with an E/B misprint.
The fourth set 4 - 661765653 - is a line from Silent Night: "ROUND yon virgin mother and child". This corresponds to the fourth puzzle's answer 'BOUND' with an R/B misprint.
So... how did you feel after you didn't get any Christmas presents last year? That's still to be determined...
