Testing network of SIM900A
To answer your questions in turn:


It is not necessary to connect to the Arduino solely for the purpose of making a GSM network connection. 
When the SIM900A is powered on, the Status LED will blink once every second.
When the GSM connection has been established the Status LED of the SIM900A will blink every 3 seconds.


Follow this guide, from Tutorial to interface GSM SIM900A with Arduino:


  Initial startup of SIM900
  
  
  Insert your SIM card to GSM module and lock it.
  Power up the SIM900A by connecting it to Arduino's 5V and GND pins
  Connect the Antenna 
  Now wait for some time (say 1 minute) and make note of the blinking rate of ‘status LED’ or ‘network LED’ (D6). 
  The GSM module will take some time to establish connection with mobile network
  Once the connection is established successfully, the status/network LED will blink continuously every 3 seconds. You may try making a call to the mobile number of the sim card inside GSM
  module. If you hear a ring back, the GSM module has successfully
  established network connection.
  
  
  Connection to Arduino
  
  The SIM900A has the following pins:
  
  
  3VR; 
  3VT; 
  5VR; 
  5VT; 
  VCC, and; 
  GND.
  
  
  For serial communication between the Arduino and SIM900A module. you
  have to connect:
  
  
  SIM900A 5VT to Arduino D9 and;
  SIM900A 5VR to Arduino D10
  
  
  Basic Communication
  
  
  To change SMS sending mode: AT+CMGF=1

mySerial.println("AT+CMGF=1");

  To read SMS in text mode: AT+CNMI=2,2,0,0,0

mySerial.println("AT+CNMI=2,2,0,0,0");

  To make a call: ATD+60XXXXXXXXX (replace X with number you want to call, change +60 to your country code)

mySerial.println("ATD+60XXXXXXXXX;"); 

  To disconnect/hangup call: ATH

mySerial.println("ATH");

  To redial: ATDL

mySerial.println("ATDL");

  To receive a phone call: ATA

mySerial.println("ATA");