How are frequency and energy related?

(1) When the frequency increases, why does the energy carried by the wave increases? (Why the energy of the wave is not only shown by the amplitude of the wave)?

A: The energy carried by the wave is conveyed by the vibrating particles on which the wave propagates. The kinetic energy of each vibrating particle (which performs simple harmonic motion) is given by (1/2).m.(2.pi.f)^2.A
where m is the mass of the particle,
f is the frequency of vibration
A is the amplitude of vibration
pi = 3.14159 .....

Thus, it is obvious that the energy carried by the wave is proportional to the square of frequency and the amplitude.
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(2) When the speed of wave increases, why does the frequency of each particle also increases? (I think when speed increases, only the wavelength changes but the frequency should remain constant)

It is NOT true. Frequency and speed are not correlated.
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(3) Why infrared radiation is emitted, is it a must for visible light to be emitted at the same time? If they emit at the same time, is it essential for the infrared radiation to be next to the red visible light?

No, most objects (as far as they are of temperature above absolute zero) emits infrared radiation, but not visible light. It is only that the temperature of an object is high enough (of order of thousand degrees) that visible light is emitted.
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(4) Do all transverse wave and longitudinal wave (no matter in straight or circular pulse) have properties of reflection, refraction, diffraction and interference?

Yes, all these are properties of waves.