What is the method of earthing the 34.5kV system?

The 34.5 kV is grounded on one end. (at switchgear's trasformer low side)

Is it solidly grounded or impedance grounded?

It is low impedence grounded.

Not sure where you are located but here's the IEEE guide:

PC62.82.2 - Guide for the Application of Insulation Coordination

Yes, it costs money; however, not using it may cost you your job, you decide.

The answer lies in your data. Since cable insulation is weak, it will protect the Lighening surges. In Insulation world, there is a saying... Weak one goes first...

Considering above, you need not install the LA for indoor MV installations...

But, yes, for out door installations, you need LA to protect the Over head line equipments.

Thanks guys!

I used IEEE C62.22 standard for doing the study, afterall.

The surge arrester have been designed in the distribution system: it is just that it needed to be on the paper to justify it to the client.