Properly installed 'push-lock' connections are 100% safe. Provided the strip gage is used and the wire is seated to the right depth they are fine. UL doesn't approve unsafe concepts.
I would buy the $5.00 circuit tester shown here, test your outlets for polarity, power and ground, then notify your landlord of deficiencies in writing. Perhaps with your next rent cheque. He cashes the cheque, you know he got the letter.
