Analysis of an Armstrong oscillator circuit
I want to understand the operation of this topology. Particularly, the
  role of C3 and C4 is not clear to me.


Lc and C3 form a supply voltage filter - in other words they try to prevent interference getting back onto the 60 volt line. This might be better explained if you gave a link to where the circuit came from. For instance, those components will also prevent noise from the 60 volts getting on to the oscillator but, the bigger picture isn't available so it's guesswork.

C4 (and the parasitic MOSFET drain source capacitance) dictate the resonant frequency of the oscillator along with L1. Loading effects due to RL on the secondary may shift this frequency slightly due to the coupling factor between the coils.


  Next, I would like to derive equations for oscillation frequency


Any derivation needs to take into account plenty of minor as well as major factors and this is best served using a simulation tool but, the main factors involve: -


L1
C4 in parallel with MOSFET capacitance
The above form a resonant circuit
C5 and the secondary coil form a series tuned circuit that feeds the load
That series tuned circuit and the coupling (0.8) will modify the oscillation frequency.


If I were to analyse it I'd probably use a sim and do an AC analysis that involved stimulating L1 and C4 (plus MOSFET capacitance) with a current source and looking where the resonant peaks might occur in the frequency spectrum for different loading values.

From that I'd consider going down the route of making a formula based on what I'd seen in the simulation.