I've an 21W T5 batten in my room at my hostel, which have a CFL like ballast with an electrodes fusing mechanism, where when one electrodes depleted emitter, it heats up and breaks, opening the ballast circuit and shutting down the lamp.
Is the filament fusing protection, can damage the ballast?

The open filament is supposed to break the oscillation (the oscillator feedback uses to be derived from the circuit current, like here via the TR1 transformer), so the ballast stops.
The only thing that may remain active is the diac "kicker" (a circuitry designed to start the oscillation), but that can not excite any large voltages and/or currents in the ballast, so can not cause any risk.

The only problem could be if either the filaments are way sturdier than they are supposed to be (so it won't break open once the emission layer is gone) or when there are too long wires between the lamp and ballast, causing the stray capacitance currents to be sufficient to keep the ballast oscillating.

But still what may have happened as the lamp was failing is quite a lot of wear on the resonant capacitor, causing it to fail. The thing is, when the lamp refuses to ignite, there is rather high voltage with high currents stressing that capacitor. Mainly when blowing the filament open takes a bit longer.
In fact this is quite a common failure mode in CFL's. But there the rating uses to really be "on the edge" and it uses to operate at rather high temperature, both accelerating the wear. In a decently designed fluorescent ballast this should not be that frequent issue.

Dor, what you mention here, is a wrong sequence of events. Really, (1) electrode activation depletes when the tube is at EOL. (2) then electron emission can not be sustained at the rated tube voltage (and normal filament temperature) and cathode drop will raise. (3) What happens next depends on the ballast.  In a classic preheat ballast (let's talk 220V part of the world) the tube will go to cycling on the stater. High frequency ballast will probably sustain the arc. (4) On a preheat ballast finally of the tube filaments will burn off. Then the tube most of the time will light up into rectification, and will work until the power is cycled. Then it will fail permanently because the starter circuit no longer can be completed and the starter is likely stuck anyway. On HF ballast, many things can happen next, depending on the schematics. EOL protection circuit may kick in on high tube voltage and shut the ballast off. Or the tube may continue to burn suffering high electrode loss and high filament temperature for some time. Then the glass may crack causing vacuum to be lost and the tube to shut off. Or the filament may burn off first, and depending on the circuit, HF generation may stop or not, or 'no tube' protection circuit may trigger then.   

Ugh... )

So it is not a ballast burning off a filament at EOL, it is a tube doing it itself if allowed to be kept under the current *beyond EOL*. Normally the ballast should be designed to survive tube EOL event, integrated ballasts which have no use after tube EOL, can be designed to fail if this is not a fire hazard.

So why in some CFLs and cheap electronic ballasts, the depleted cathode being overloaded and when it is breaks, the lamp shuts down? https://www.lighting-gallery.net/gallery/displayimage.php?pos=-130190

Once the lamp reaches EOL by consuming all the available emission material, the cathode fall rises. This causes the voltage to rise and that in turn causes higher current through the resonant capacitor as the LC is boosting the voltage for the lamp (the inverter itself generates just about 130V, so anything higher must come from a resonant boost).
That, along with the elevated cathode losses (because of the higher cathode fall) causes the cathode filament to break.
Because the filaments are in series with the LC resonator circuit, the circuit becomes unable to boost the voltage, causing the discharge to extinguish.
At that point the current stops flowing. And because with the simple "CFL-like" circuits the feed back signal to drive the transistors is derived from the circuit current, breaking this circuit means the oscillator stops.

So far that is how the EOL and after events are supposed to evolve (on those simple 2-transistor selfoscillating "CFL-like" ballasts), in order for the ballast to survive and no "glass shower" nor fire coming from the lamp EOL.

There are few from things what may go wrong:
- The filament is way sturdier, so it takes longer for it to break, so the high current/high voltage condition remains for too long killing the ballast. The cause could be bad tube design, but as well some gas contamination/leak (that caused by e.g. a seal crack EOL) causing excessive filament cooling.
- Even the short time HV/high current is enough to become the last "nail in the coffin" for the ballast components
- The cold cathode operation warms up the cathode assemblies so the discharge remains, meltingthe lead wire and the glass. This then may result to glass tube breaking (and so "glass shower"), or overheating the end assemblies so the socket melts (so either fire risk, or loosing the tube letting it fall, so again the "glass shower")
- As the filament breaks, a discharge arces over and sustain there, leading to end overheating (with the same consequence as above)

Because of the past-EOL tube behavior is not that well predictable (loss of emission material is not the only EOL mode), good ballast do feature circuit designed to detect the EOL and shut down the ballast in a more controlled way, not relying for the lamp parts (mainly the filament) to disintegrate...

Thanks.

@Medved: Good point about increasing lamp voltage also increasing capacitor current, that way making filament temperature increase even further. Never thought about this. By the way, resonant capacitor is not always connected across the 'cold' ends of filament in an electronic ballast. Notably, in an old oddball Hungarian-made GE CFL ballast, that one with a single choke/gate transformer magnetic and two complementary SMD MOSFETs as switches, resonant cap was connected across the hot ends of filaments, so its current did not flow through the tube! Only a starting PTC with a second series capacitor was connected across the cold ends.

The resonant cap split between hot vs cold ends is used when the tube needs voltage boost during normal burn (arc voltage is too close or above the inverter output). In that condition there is a significant current flowing all the time through the resonant capacitor (comparable to lamp arc current; normally it is about 1/3). There are few ways to deal with that:
One of the most frequent on integrated CFLs is just to design the filaments for higher current and take that heating current into cathode filament thermal budget. In that way the circuit remains simple, include somewhat reasonable EOL cutout for a single use ballast. The drawback is the need for a specially tailored design of the tube, but not a big deal with a selfballasted CFL. Of course, it becomes problematic for virtually any standardized tube.
Other way I have seen is to bypass filaments with diodes. That way the heating current is reduced to about 70%, but once the filament breaks, the flashover discharge is more likely. But because diodes are small and cheap, it is used, mainly on 120V CFLs (to avoid the need for a doubler), where the flashover is less likely.
Other option is to split the resonant capacitor to half and connect each from hot end on one side to the cold end on the other tube side. This reduces the residual filament current to 50%, but needs both filaments to break for it to shut down. But once one breaks, the resonant circuit changes its impedance, yielding lower current so power, so often able to survive tye extra few hours till the other filament breaks too. But in that condition it means higher load on the remaining capacitor. But to the remaining filament as well, making it more likely to break too.

And then there are two other options, able to make really high voltage boost (2x or more):
One is you are describing, one capacitor connected directly on the coil and only the other part passing the filament current.
Other has all the resonant capacitor connected directly and the filaments get power in a separate way, either via a separating transformer or auxiliary windings on the main ballasting inductor. Thus later is used mainly with dimmable ballasts, as it is not that difficult to make the filament voltage tracking what it needs to maintain the temperature when dimmed down.
These two have the drawback of principally continuing operation even when the lamp is not there, so it needs additional measures to keep it safe.
One option (for the first one) is to make the oscillator frequency not that much dependent on the ballasting tank resonance. That way once the part of the capacitor connected via the filaments is gone, the remaining LCis detuned so much it does not generate the dangerous voltages/currents anymore. The drawback is the circuit sensitivity to the values of the power passives, which allows onlt a limited wear of them, so shorter useful life.
And other method is an explicite dedicated EOL detection/shut down circuit. This is tye most complex, so used very rarely with selfoscillating circuits.
On the other hand this complexity is rather easy and cheap to integrate into a ballast control IC (there even 1000 transistors extra are way cheaper than rating any power component with just even 10% extra margin), so practically all ballast ICs do feature some form of that. But because those are more expensive than the simple selfoscillating circuits, it is used only when the ballast really needs to last long (ballasts for standard fluorescents,...) or if it needs some complex functionality anyway (e.g. precise programmed start for high cycle life or dimming or so).