When input1 goes high, the voltage at the output of your NOR gate drops because Q2 turns on. This voltage couples back through the collector-base capacitance of Q1, causing the spike on input2. The problem is caused by the fact that you are using two nuclear bombs (2N3055, which has huge junction capacitance) to kill a mosquito. Change the transistors to any small-signal part (e.g. 2N3904), and the problem will go away. You don't need R3 and R4. R1, R2, and R9 can be at least 10 times larger. Your base resistors can be 10 to 20 times the parallel resistance of R1 || R2 || R9. Also, your timing resistor (R8) can be any value up to 100k, allowing you to use a smaller value for your timing capacitor.
The spike will still be there to some extent. You can clip it off with a 1N4148 from base to GND, anode to GND, if it bothers you. You should have one on each base, although I wouldn't worry about it. Base-emitter breakdown voltage on most transistors is at least 5 volts.