GSM ZTE MG2639 module freezes after checking SIM

Okay, I was heading down dead ends there. The vendor just confirmed that there is a fault with the SIM card holder on the board.

Solution is to reflash SIM900A with SIM900 software and it works like a charm.

Here is the page that shows the solution.

Flashing is no trickery, just a right sequence of operations, there is detailed instruction for arduino-like approach, my way required just a few major steps:

1. Power up your SIM900A module to respond to AT commands in auto-baud mode (AT+IPR=0). It means you must wire powerlines, do power-on sequence with PWRKEY etc.
2. Setup flashing tool — connect it to SIM900A, set default speed of 460800 baud, choose firmware to flash from unpacked RAR file, select “don’t check file name” check-box and press “start download” button to see “Power On/Reset Target” message.
3. Short NRESET (pin 16) to GND for a moment. Right after it reboots flashing tool will start pushing data.
4. When flashing ends with “Download complete” power off SIM900A module. Note that reset will not work, disconnect power line and start power-on sequence once again.
5. Voila! It works now as expected. PIN accepted, logged in to network and test text message successfully sent to my smartphone.

Thanks everyone for helping out!