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Can you find the message hidden within this grid of puzzles?

enter image description here

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    $\begingroup$ Note there's no rush to accept an answer until you are fully satisfied all points are explained... $\endgroup$
    – Stiv
    Mar 27 '20 at 15:36
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Going through the sixteen clues (four or five I still couldn't solve completely):

  1. $T$ (the letters so far are the 4th, 8th, 12th, 16th, so we need the 20th)

  2. $O$ (going from top to bottom in each column, we should add 4 then 6 then 8, matching the arithmetic progression in the top row)

  3. $D$ (arithmetic with Roman numerals)

  4. Answer should be $A$ (see below), but I'm not sure why. Going from the first to second row, we seem to be jumping four letters ahead, but that would mean $W$ and the third row doesn't factor in. In the first column we have $3,7,25$ and in the second column $11,15,17$, but I can't see the common thread there. Edit: see Stiv's answer.

  5. $Y$ (the letters are the multiples of 5 through the alphabet, 25 missing)

  6. Answer should be $S$ (see below), but I'm not sure why. Alternating sectors have OOU and J?Y; opposite pairs are JO (10,15) and OY (15,25).

  7. Answer should be $M$ (see below), but I'm not sure why. The first two are simply "jump two letters back in the alphabet), but that doesn't match the third one. Edit: see Stiv's answer, although I don't like this clue since there are different possible ways of writing the name of each letter.

  8. $Y$ (Atbash cipher)

  9. $B$ (taking alternate sectors we have HIJ and A?C)

  10. $I$ (the letters are the multiples of 3 through the alphabet, 9 missing)

  11. $R$ (the letters are the multiples of 6 through the alphabet, 9 missing)

  12. Answer should be $T$ (see below), but I'm not sure why. There are some words we should figure out, but the only thing coming to my mind for the first two lines is OWL WHO ... Edit: see Stiv's answer.

  13. $H$ (the number for each letter is the sum of the numbers for the two below it)

  14. $C$, probably (the letters are AB? and OPQR). Answer should be $D$ (see below), but I'm not sure why. Edit: see Stiv's answer.

  15. $A$ (in each trio, starting from top left and going clockwise we have alternate letters, J(K)L(M)N and M(N)O(P)Q)

  16. $Y$ (in each diamond, the pairs of opposite letters have the same sum, 27 below and 26 above)

Note that most of these are actually number puzzles in disguise :-)

Overall:

TOD?Y??YBIR?HCAY

So I should say:

Happy birthday! (and then figure out the missing clues)

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Independently of @Randal'Thor's answer (which was posted in its original form just a few minutes before this one), I have explanations for all 16 puzzles (15 solid, 1 less so), and must say from the off:

Happy birthday to you!

The puzzles resolve as follows (I am least sure on #6):

1 = T (sequence – 4 letters along each time);
2 = O (middle letters of each row are halfway between those on left and right);
3 = D (500 in Roman numerals, as per the result of the Roman mathematics);
4 = A (starting top left, then circling round the outside each letter is 8 further on through the alphabet than the previous, beginning again from A after Z);
5 = Y (the 5-sided star is surrounded by all letters whose A1Z26 value is divisible by 5 – the missing letter is #25, Y);
6 = S (DUBIOUS: In the absence of an obvious pattern or sequence, I suggest that 'S' is the only letter which can be added to the others and anagrammed to form a real word in English: JOYOUS);
7 = M (the letter to the right of the equals sign is the last letter of the first when spelled out in full);
8 = Y (letters equidistant from opposite ends of the alphabet);
9 = B (A1Z26 values of opposing letters sum to 11);
10 = I (the 3-sided triangle is surrounded by all letters whose A1Z26 value is divisible by 3 – the missing letter is #9, I);
11 = R (the 6-sided hexagon contains all letters whose A1Z26 value is divisible by 6 – the missing letter is #18, R);
12 = T (words are the first four numbers: ONE, TWO, THREE, FOUR – required letter is first of the third word, i.e. THREE);
13 = H (sum of A1Z26 values of two consecutive letters equal the value of the letter above them);
14 = D (all letters in the circle contain ‘closed counters’, i.e. can be coloured in – the one missing is ‘D’);
15 = A (each triple consists of 3 letters, sequentially 2 letters along the alphabet, beginning top left);
16 = Y (A1Z26 values of letters in diagonals of each diamond are equal – first diamond’s diagonals must sum to 26)

Altogether, these spell out:

TODAY'S MY BIRTHDAY!

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