I am not sure I catch all of your questions.
1) Voltage output versus input: Check the datasheet. The 555 you are using is not CMOS and cannot swing its output to the power rails. Thus, the max output voltage is approximately Vcc-1.7V at 100 mA. That voltage drop increases slightly at higher currents. That is all explained under "Electrical Characteristics." For the LM555, the output voltages are described on page 4 of its datasheet.
2) Frequency: The 555 is a robust device and is relatively stable. However, the frequency of this circuit will vary with the supply voltage. (Note, there are diode voltage drops introduced, not just resistors.) What voltage do you intend to use? I did the spice model at 5V for the reasons stated in that post. Now it seems you have tried some other voltages. Pick just one voltage that you want to use. Let's get it working at that voltage. Then, if you want to change the voltage, you will have a starting point to know what will need to be adjusted.
3) The higher frequency oscillator in the asc file you attached is oscillating at 46 KHz in simulation. That is probably due to the voltage used. See #2, above. When you reference a specific frequency, is that measured on a prototype or from the simulation?
John