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_{This is an entry into the Fortnightly Topic Challenge #41.}

The 100th anniversary of the end of World War I just passed a few days ago! In memory of this, in this puzzle we're going to reconstruct an ancient war, as described by a veteran of the war itself. The grid below will serve as our map of the battleground.

(1) The war took place between two sides. Both sides claim they gained from the war, and each clue has also gained a letter. Taking the extraneous letters in order of the clues will show you what each side gained.

(2) The armies communicated by a simple code: for a clue numbered n, shift the clue's n-th alphanumeric character down by n. This will show you the story-teller's only source of comfort during the war, as well as indicate what he lost during the war.

(3) Once the battlefield was full, each side requisitioned some back-ups, which caused the inhabitants of exactly five cells to multiply accordingly. After performing this transformation, you'll be able to see the aftermath of the war, and adding one more upright line will show you who won!

Across
1. Uninitiated, done receiving student's cheer (3)
4. Scam transmission addressed originally to disco (3)
5. 110 sheltering Uno scientist Brian (3)

Down
1. User's self-made persona consumes top gin fizz -- in short, it's only natural! (2 1)
2. School, after mid-term: Seth up to John (3)
3. Life-tender with two of the same chromosome, briefly! (2'1)

That is a very impressive amount of content to fit in such a tiny cryptic. If you haven't had the time to solve it yourself, I recommend stopping here until you have the chance. Let's start with the clues.

Across
1. Uninitiated, done receiving student's cheer (3)

The extra letter is the N in "doNe". The clue should be "Uninitiated, doe receiving student's cheer (3)".
OLE = "cheer"; "uninitiated doe" = (-d)OE containing "student" = L

4. Scam transmission addressed originally to disco (3)

The extra letter is the O in "discO". The clue should be "Scam transmission addressed originally to disc (3)".
FOX = "Scam"; "transmission" = FAX, with "addressed originally" = A_ exchanged to "disc" = O

5. 110 sheltering Uno scientist Brian (3)

The extra letter is the U in "Uno". The clue should be "110 sheltering no scientist Brian (3)".
COX = "scientist Brian"; "110" = CX containing "no" = 0 = O

Down
1. User's self-made persona consumes top gin fizz -- in short, it's only natural! (2 1)

The extra letter is the G in "Gin". The clue should be "User's self-made persona consumes top in fizz -- in short, it's only natural! (2 1)".
OF C = "in short, it's only natural", text short for 'Of Course'; "User's self-made persona" = OC (original character) containing "top in fizz" = F_

2. School, after mid-term: Seth up to John (3)

The extra letter is the H in "setH". The clue should be "School, after mid-term: Set up to John (3)".
LOO = "John" as in toilet; "School, after mid-term" = (-sch)OOL, "set up" = reversed

3. Life-tender with two of the same chromosome, briefly! (2'1)

The extra letter is the T in "life-Tender". The clue should be "Life-ender with two of the same chromosome, briefly! (2'1)".
EX'X &lit.; the definition is a female executor, which is an executrix, abbreviated to EX'X; the wordplay is "Life-ender" = _E, with "two of the same chromosome, briefly" = X+X

This makes the filled grid this:

OLE
FOX
COX

(1) What did each side gain?

The war was ultimately fruitless; each side gained NOUGHT, seen in the extra clue letters already indicated.

(2) What was the story-teller's only source of comfort during the war? What did he lose during the war?

First letter of 1-Across clue = U; shifted back by one = T

Fourth letter of 4-Across clue = M; shifted back by four = I

Fifth alpha-numeric character of 5-Across clue = H; shifted back by five = C

First letter of 1-Down clue = U; shifted back by one = T

Second letter of 2-Down clue = C; shifted back by two = A

Third letter of 3-Down clue = F; shifted back by three = C

The only source of comfort was TIC-TAC-TOE, but the story teller lost their TOE.

(3) How do the arriving reinforcements change the battlefield?

The five cells with the numbers 1,2,3,4,5 in them are the ones that are multiplied.

Take the letter inside and convert with A1Z26. Multiply by the number corresponding to that cell, and convert back. (e.g. the cell with 3 has an E inside. E is the 5th letter of the alphabet. Multiply 5 by 3 to get 15, and replace with the 15th letter of the alphabet, an O.)

(3) How does the battlefield appear after back-ups arrive?

This makes the filled grid this:
OXO
XOX
OOX

(3) What is the aftermath of the war? Who won?

This filled grid is a Tic-Tac-Toe game. Adding an 'upright' line (that is, a line that goes up-and-right) from the bottom-left corner to the top-right corner, shows that the game was won by the O side.
. /
. OXO
. XOX
. OOX
./