Frequency of a Tuning Fork

Nope. The frequency of the tuning fork doesn't depend on the force with which it's struck. Every tuning fork has its own desired frequency. And hence, people usually say, "Hey - take that fork, the one with 432 Hz (an example) on it"...

The oscillations usually get damped out after some period. But, the frequency still remains the same... Usually, the frequency of a tuning fork depends only on the property of its material.

If you have a look at the Wiki article on frequency:-

$$F \alpha \frac{1}{l^2}\sqrt{\frac{EI}{\rho A}}$$

This clearly shows that the parameters are all the properties of the material on which the fork is made...

The frequencies of vibration of a tuning fork can be affected by the force used to strike it. Depending upon the magnitude, direction, and the number and location of the points of application, vibrational modes other than the fundamental mode could be excited. Each of these vibrational modes have a different frequency. The mode frequencies are characteristic, but not all necessarily need to be excited every time. For a tuning fork, the fundamental mode, the clang mode, the asymmetric in-plane mode, the out of plane bending mode, and the asymmetric out of plane modes are shown here as animations.

The frequency engraved on the tuning fork is for the fundamental mode. As a device designed to act as pitch reference, the other modes (of a high-quality fork) should not be easy to excite strongly, and should damp out quickly. But you should be able affect how much the auxiliary modes are excited by how you strike the fork.

I believe that the force doesn't affect the frequency of the sound generated, based on how guitar strings work. Addressing each point:

• You have different strings of different densities which give different tones, so yes, tine stiffness is a factor.
• You press the string at different positions to give different tones, so yes as well for the length.
• I never noticed any change in the tone whether the strings are stuck stronger or weaker. The volume is different, not the tone.
• If you really mean the frequency of the sound the fork produces, then I would say yes based on how people can use carbon dioxide to make their voice sound higher pitched.
• Since density of air is immediately related to temperature, I would say yes to this as well.

EDIT: Since it seems that the question is not about the sound the tuning fork produces, the last two options are not valid.