Unclear block diagram of Hamamatsu's Hybrid Photo Detector
Are you familiar with photomultiplier tubes?  Hamamatsu has a very detailed handbook that reviews their operation.  If you're going to do circuit level design with a hybrid PMT, you should probably take a quick look at it:
https://www.hamamatsu.com/resources/pdf/etd/PMT_handbook_v4E.pdf

What exactly the two 2.5G Ohm of resistors doing there? It seems as the HV supplied will be directly fed to the photocathode. They look like potential divider but not really doing their voltage divider purpose here.

These hybrid detectors are photomultiplier vacuum tubes pointed at avalanche photodiodes.  The avalanche diode is biased at some relatively low voltage (probably 10s of volts).  The Photomultiplier tube is biased at a very high voltage (probably 1000s of volts) split across multiple plates that generate the electron gain.  The voltage divider is for the internal plates so that they end up at an intermediate voltage between the anode and the cathode.

Why is the other end of the series resistors connected to the &quot;casing&quot; that enclosed the photocathode and avalanche diode (in solid line)? I can't see the purpose of that.

The casing is the vacuum tube.  The middle wire goes into the vacuum tube and powers the internal plates.

What are the resemblance of vertical dash line that the capacitor is connected to? The ground of the coaxial wire that connects to the High-Speed Amplifier?

That is signal ground (or maybe better called return since it may not be attached to actual ground).  The capacitor is a conventional photodiode bias filter.  It provides charge to the APD so that when it fires the supply voltage doesn't transiently drop, so it is just a capacitor from the bias line to the bias ground.  Think of it as decoupling an IC.
As an aside, in my experience the main use of a hybrid detector is fast TCSPC due to their extremely repeatable single photon response.  For other uses there are usually better options.  If you're doing TCSPC, you probably buy the hardware from the OEM and plug in the hybrid detector without worrying too much about how its biased internally.