Some characters are not transmitted correctly from mega32 to ESP8266
Summary:  Your symptoms fit with marginal UART bit timing errors. Use UBRRL = 0x67; instead of UBRRL = 0x68; for 9600 baud (really bits per second) UART from a 16MHz CPU clock, as recommended by Atmel.

Details:
The fact that both your ESP8266 module and your logic analyser's own UART decoding, are failing to correctly decode some of your "9600 baud" ATmega32 UART output, is a clear indication that the ATmega32 is the likely source of the problem.

Then, the fact that there is sometimes a difference in interpretation between those two receivers (e.g. logic analyser doesn't decode '/r' or '/n' but the ESP8266 accepts the command and so it must have decoded one or both characters), shows that the problem is marginal.


  For some reason mega32 cannot accurately transmit all the characters and I cannot understand why this happens.


You are correct, and well done for using a logic analyser.  You would have found the problem yourself, if you used the analyser to get more precise timings of each bit - it is your bit rate (baud rate) which is slightly wrong, because you have used:

UBRRL = 0x68;


Actually, for an ATmega32 with 16MHz clock, the correct value is:

UBRRL = 0x67;


The slightly incorrect data rate is what affects different UART receivers differently (as you found yourself).

As I said, you just needed to progress a little further by checking the bit timings (an oscilloscope is typically used for this, but I expect your logic analyser can show the necessary bit timings too) and have the confidence that you were "on the right track" with your analysis of the observed results, and I'm sure you would have found the answer yourself :-)



Edited to add:

See the calculation here: Stack Overflow: What is the function of UBRRH and UBRRL registers in atmega32?

See the Atmel Application Note AVR306: Using the AVR USART on tinyAVR and megaAVR devices - Table 3-5 on page 7 shows that the correct UBRR value for 9600 baud with a 16MHz clock is 103 decimal = 0x67.