Doesn't Arduino reports DSR (Data Set Ready) correctly? I was told, that when computer opens port, it sends DTR (Data Terminal Ready) signal to Arduino, which initiates Arduino reboot.

This looks like part of DTR / DSR flow control. According to this, in response to DTR, Arduino should no only reboot, but also raise DSR (Data Set Ready) signal when it is ready to receive data

And computer should wait for it.

Does it happen so?
No, it does not, and can not for the simple reason that there is no DSR line connected to the processor.

Generally host software / serial driver configuration ignores DSR.

On the boards with an 8u2/16u2 the custom firmware therein can report whatever virtual state of the signal it is programmed to. On those with hardware USB-serial they will report however the pin is hard wired, or whatever value it is pulled to or floating at.  

None of these (possibly excepting a Leonardo type board or an ARM one with native USB) reflect anything about the state of the target processor, so software hoping to talk to an Arduino should ignore the DSR signal.