Connection problem with Arduino Mega 2560 and GSM shield ( official M10)
I have done this, and had (some limited) success.

A couple of your connections are wrong;
Connect both the pin 7s on the shield and MEGA, not the pin 4s. Pin 7 is the reset pin - I think that GSM.begin resets the shield before doing anything. This might help with Q1.

Make sure that you use the regulated 5V output on the MEGA, and not the Vin to power your shield. Vin is whatever voltage is connected to the MEGA power socket, and could damage your shield. I personally have not noticed any difference when powering the MEGA+Shield from USB or external 12V supply.

I had problems with interference when I used wires to connect shield to MEGA. The stacking seems to have reduced this problem, but not eliminated it by any means...

The GSM Web server is unlikely to work except maybe from within your SIM provider's own network. Some of the limitations are mentioned here;
https://www.arduino.cc/en/Tutorial/GSMToolsTestWebServer