What AT Commands should I use to read RxLevel, RXQuality, CellID, LocationAreaCode, RSCP, EcN0, CQI?
AT+CREG? gives CellId and LAC
+CSQ gives RSSI in dBm and RXQUAL
See 3GPP TS 27.007 for standard 3GPP AT commands, available from www.3gpp.org. Your device probably supports some manufacturer-specific AT commands as well.
Is it possible to read Level 3 signalization by using AT Commands?
No, you can't read RRC messages, only selected parameters from them. You can read the layer 3 signalling messages using Qualcomm diagnostic tools, but you have to have an agreement with Qualcomm to get these.
Is there better Android Qualcomm API to get these informations? What is Gobi / QMI interface and can it be used for reading Network data?
Gobi/QMI is the Qualcomm interface. There is also information about AT commands supported by Qualcomm devices at Qualcomm documentation