So, if INTERRUPT is not use, NeoHWSerial does not have the advantage of using it?
NeoHWSerial is HardwareSerial, with one addition: attachInterrupt. I started with the Arduino core source code and made this modification. If you don't need attachInterrupt, you might as well use the built-in HardwareSerial (i.e., Serial, Serial1, etc.).
I thought OBD would read on serial(0) and then (at the same time) print it on USB. If this will work? Otherwise, I can not read in Serial Monitor Window.
Yes, that will send debug messages to the OBD device as well as the Serial Monitor window. Because the device requires a certain AT command format, it should ignore your debug messages... unless they start with "AT". Then it tries to interpret your debug messages. 
Don't do that.
You will also see the AT commands in the Serial Monitor window. But maybe you should connect the OBD device to Serial1, Serial2 or Serial3, like I suggested earlier.
You could put the GPS on Serial, if you want. Again, Serial1, Serial2 or Serial3 would be more convenient, because you wouldn't have to disconnect pin 0 everytime you want to upload a new sketch.
It works. Is this the most power saving mode? I don't need hot start, I just want to turn off all I can when car is turned off.
Yes, this is the lowest power mode for the ublox devices: 35uA is a very small drain. Other system components probably use more than this (e.g., regulators' quiescent currents). You'll have to measure the actual usage of your system.
Be sure to read reply #5 and reply #20 in that other thread.
Do you have any idea how to wake the whole system from sleep when I start the car? If is there any software "tweak"? I want to avoid additional cables/connections.
No clue. How would your system know the car is "on"?
Maybe a tilt switch or vibration/sound sensor hooked to a Pin Change Interrupt? Sensing a voltage on something must be done very carefully, because automotive power if very noisy, even dangerous to the components.
The Watch Dog Timer can be used to periodically wake up, but it increases the sleep current.