First of all one should have clear the difference between classical electromagnetic theory, which completely describes synchrotron radiation with the potentials generated by a classical moving charged particle, and the quantum frame for EM which consists of photons building up the classical radiation.
What frequency will the radiation of the second animation be?
Synchrotron radiation has a spectrum of frequencies as seen in the lecture referenced above:
It is a continuum of photon energies emitted. At the quantum mechanical level, it is the probabilistic interaction of the charged particle with the field that is accelerating it.
The lower photon is off mass shell giving the energy, from the accelerating field, for the radiation of the real photon (upper).
If a charged particle is then hit by the radiation emitted by the second animation, would it just feel a force in a single direction, rather than an oscillatory force?
It depends on the size of the charged particle. At the quantum level it is only energies and spins that are exchanged. The single photon does not oscillate, only its wave function whose complex conjugate squared will give the probability density for the interaction to occur.
In terms of the actual photons, it would seem the first animation emits only photons with the same frequency as the particle's oscillation.
The dipole emits classical EM light with zillions of photons of energy = $h \ \nu$, the frequency, because it needs the ~1023 atoms of the dipole antenna. A single quantum mechanical particle is not a dipole (it can radiate as a dipole in an imposed field which has alternating frequency), it has to be imaged by Feynman diagrams with individual photon probabilistic radiation.
so would it would emit the exact same photons for that period?
The emission will follow the probability distribution for the solution of the above feynman diagram.
Best Answer
This is my personal favorite one of those (from MTW's "Gravitation").
For an animation, see, for example this Java applet.