Krupski:
I can't believe this is so confusing!
Me neither. 
Now, If I open a serial port at 19200 baud and "listen" to it, then clicking "Burn Bootloader" should send me the various command bytes that are used to program the AVR chip.
Oddly enough, I can reproduce this. It might have been simpler to post your "tiny sketch" rather than all this stuff about your "own" Arduino as ISP, with LCD screens etc. which makes one wonder if you have done something strange. But, no matter.
With this test sketch:
const byte LEDin = 6;
const byte LEDout = 7;
void setup ()
{
Serial.begin (19200);
pinMode (LEDin, OUTPUT);
digitalWrite (LEDin, LOW);
pinMode (LEDout, OUTPUT);
digitalWrite (LEDout, LOW);
} // end of setup
void loop ()
{
if (Serial.available ())
{
digitalWrite (LEDin, HIGH);
byte c = Serial.read ();
digitalWrite (LEDin, LOW);
digitalWrite (LEDout, HIGH);
Serial.write (c);
digitalWrite (LEDout, LOW);
}
} // end of loop
That should read from Serial, and write whatever it gets back out. Plus some debugging LED flashes.
Results:

The 3 blips with a red box around are the '0' bytes sent by avrdude, like this:
avrdude: Version 5.11, compiled on Sep 2 2011 at 18:52:52
Copyright (c) 2000-2005 Brian Dean, http://www.bdmicro.com/
Copyright (c) 2007-2009 Joerg Wunsch
System wide configuration file is "/Applications/Arduino_1.0.3.app/Contents/Resources/Java/hardware/tools/avr/etc/avrdude.conf"
User configuration file is "/Users/nick/.avrduderc"
User configuration file does not exist or is not a regular file, skipping
Using Port : /dev/cu.usbmodemfa131
Using Programmer : stk500v1
Overriding Baud Rate : 19200
avrdude: Send: 0 [30] [20]
avrdude: Send: 0 [30] [20]
avrdude: Send: 0 [30] [20]
avrdude: ser_recv(): programmer is not responding
avrdude: stk500_recv(): programmer is not responding
avrdude done. Thank you.
The one on the left and one on the right are 0xFF.
So you appear to be right. The Mega is behaving very strangely. It gets the data, because the "LEDin" LED goes high. However after that it just seems to hang. Although LEDout goes high as well. However LEDin doesn't go low. Nor does anything get sent.
LEDin is high for 6.6 seconds!
If I load the ASCII map (example) onto the same board, it works:
ASCII Table ~ Character Map
!, dec: 33, hex: 21, oct: 41, bin: 100001
", dec: 34, hex: 22, oct: 42, bin: 100010
#, dec: 35, hex: 23, oct: 43, bin: 100011
$, dec: 36, hex: 24, oct: 44, bin: 100100
%, dec: 37, hex: 25, oct: 45, bin: 100101
&, dec: 38, hex: 26, oct: 46, bin: 100110
', dec: 39, hex: 27, oct: 47, bin: 100111
(, dec: 40, hex: 28, oct: 50, bin: 101000
), dec: 41, hex: 29, oct: 51, bin: 101001
*, dec: 42, hex: 2A, oct: 52, bin: 101010
+, dec: 43, hex: 2B, oct: 53, bin: 101011
,, dec: 44, hex: 2C, oct: 54, bin: 101100
-, dec: 45, hex: 2D, oct: 55, bin: 101101
., dec: 46, hex: 2E, oct: 56, bin: 101110
/, dec: 47, hex: 2F, oct: 57, bin: 101111
0, dec: 48, hex: 30, oct: 60, bin: 110000
1, dec: 49, hex: 31, oct: 61, bin: 110001
2, dec: 50, hex: 32, oct: 62, bin: 110010
3, dec: 51, hex: 33, oct: 63, bin: 110011
4, dec: 52, hex: 34, oct: 64, bin: 110100
5, dec: 53, hex: 35, oct: 65, bin: 110101
6, dec: 54, hex: 36, oct: 66, bin: 110110
7, dec: 55, hex: 37, oct: 67, bin: 110111
8, dec: 56, hex: 38, oct: 70, bin: 111000
9, dec: 57, hex: 39, oct: 71, bin: 111001
:, dec: 58, hex: 3A, oct: 72, bin: 111010
;, dec: 59, hex: 3B, oct: 73, bin: 111011
<, dec: 60, hex: 3C, oct: 74, bin: 111100
=, dec: 61, hex: 3D, oct: 75, bin: 111101
>, dec: 62, hex: 3E, oct: 76, bin: 111110
?, dec: 63, hex: 3F, oct: 77, bin: 111111
@, dec: 64, hex: 40, oct: 100, bin: 1000000
A, dec: 65, hex: 41, oct: 101, bin: 1000001
B, dec: 66, hex: 42, oct: 102, bin: 1000010
C, dec: 67, hex: 43, oct: 103, bin: 1000011
D, dec: 68, hex: 44, oct: 104, bin: 1000100
E, dec: 69, hex: 45, oct: 105, bin: 1000101
F, dec: 70, hex: 46, oct: 106, bin: 1000110
G, dec: 71, hex: 47, oct: 107, bin: 1000111
H, dec: 72, hex: 48, oct: 110, bin: 1001000
I, dec: 73, hex: 49, oct: 111, bin: 1001001
J, dec: 74, hex: 4A, oct: 112, bin: 1001010
K, dec: 75, hex: 4B, oct: 113, bin: 1001011
L, dec: 76, hex: 4C, oct: 114, bin: 1001100
M, dec: 77, hex: 4D, oct: 115, bin: 1001101
N, dec: 78, hex: 4E, oct: 116, bin: 1001110
O, dec: 79, hex: 4F, oct: 117, bin: 1001111
P, dec: 80, hex: 50, oct: 120, bin: 1010000
Q, dec: 81, hex: 51, oct: 121, bin: 1010001
R, dec: 82, hex: 52, oct: 122, bin: 1010010
S, dec: 83, hex: 53, oct: 123, bin: 1010011
T, dec: 84, hex: 54, oct: 124, bin: 1010100
U, dec: 85, hex: 55, oct: 125, bin: 1010101
V, dec: 86, hex: 56, oct: 126, bin: 1010110
W, dec: 87, hex: 57, oct: 127, bin: 1010111
X, dec: 88, hex: 58, oct: 130, bin: 1011000
Y, dec: 89, hex: 59, oct: 131, bin: 1011001
Z, dec: 90, hex: 5A, oct: 132, bin: 1011010
[, dec: 91, hex: 5B, oct: 133, bin: 1011011
\, dec: 92, hex: 5C, oct: 134, bin: 1011100
], dec: 93, hex: 5D, oct: 135, bin: 1011101
^, dec: 94, hex: 5E, oct: 136, bin: 1011110
_, dec: 95, hex: 5F, oct: 137, bin: 1011111
`, dec: 96, hex: 60, oct: 140, bin: 1100000
a, dec: 97, hex: 61, oct: 141, bin: 1100001
b, dec: 98, hex: 62, oct: 142, bin: 1100010
c, dec: 99, hex: 63, oct: 143, bin: 1100011
d, dec: 100, hex: 64, oct: 144, bin: 1100100
e, dec: 101, hex: 65, oct: 145, bin: 1100101
f, dec: 102, hex: 66, oct: 146, bin: 1100110
g, dec: 103, hex: 67, oct: 147, bin: 1100111
h, dec: 104, hex: 68, oct: 150, bin: 1101000
i, dec: 105, hex: 69, oct: 151, bin: 1101001
j, dec: 106, hex: 6A, oct: 152, bin: 1101010
k, dec: 107, hex: 6B, oct: 153, bin: 1101011
l, dec: 108, hex: 6C, oct: 154, bin: 1101100
m, dec: 109, hex: 6D, oct: 155, bin: 1101101
n, dec: 110, hex: 6E, oct: 156, bin: 1101110
o, dec: 111, hex: 6F, oct: 157, bin: 1101111
p, dec: 112, hex: 70, oct: 160, bin: 1110000
q, dec: 113, hex: 71, oct: 161, bin: 1110001
r, dec: 114, hex: 72, oct: 162, bin: 1110010
s, dec: 115, hex: 73, oct: 163, bin: 1110011
t, dec: 116, hex: 74, oct: 164, bin: 1110100
u, dec: 117, hex: 75, oct: 165, bin: 1110101
v, dec: 118, hex: 76, oct: 166, bin: 1110110
w, dec: 119, hex: 77, oct: 167, bin: 1110111
x, dec: 120, hex: 78, oct: 170, bin: 1111000
y, dec: 121, hex: 79, oct: 171, bin: 1111001
z, dec: 122, hex: 7A, oct: 172, bin: 1111010
{, dec: 123, hex: 7B, oct: 173, bin: 1111011
|, dec: 124, hex: 7C, oct: 174, bin: 1111100
}, dec: 125, hex: 7D, oct: 175, bin: 1111101
~, dec: 126, hex: 7E, oct: 176, bin: 1111110
Using the above test sketch (with the LEDs) and simply typing "abcdef" into the serial monitor, it works as expected:

I don't know what to make of this, except that maybe the Serial monitor initializes the USB chip (eg. by sending RTS low) and that avrdude doesn't. I can't think of any other explanation. And even that doesn't explain why LEDin stays high for so long.