Is this the minimum power factor that the generator can handle? or Is it the power factor that the generator should work at?
It is the minimum power factor for full-load operation. It can probably handle a lower power factor with reduced load.
I mean If I connected a load of 0.95 power factor. Is it better than a load of 0.8 power factor?
It is better in terms of reducing the losses in the generator, but it won't let you deliver more than rated watts.
If I would like to design a capacitor that corrects power factor. Should I correct the power factor of the load to be 0.8 or 1 ?
Correcting to 0.8 would be good. Correcting higher might not provide much benefit.
Does the generator have an internal capacitor to correct power factor? so that when I connect a load of 0.8 power factor, The generator will see a power factor of 1.
A wound field synchronous generator will supply reactive current up to the capability of its excitation control system. That isn't quite the same thing as correcting the power factor and it doesn't involve an actual capacitor, but it does make the generator behave like it has an internal capacitor.
More about generator ratings
For an engine-generator set, the KVA rating is basically determined by the maximum current. The KW rating is determined by the power rating of the engine. A generator set rated 5 KVA, 0.8 pf, the engine would be capable of generating 4 KW. Operating at 5 KVA and 4 KW would be operating at the maximum capability of both the engine and generator. If the power is 4 KW but the pf is less than 0.8, the KVA would exceed the generator rating. If the generator is operating at 5 KVA and the pf is higher than 0.8, the KW would exceed the engine rating.
No matter how the relative ratings of the generator and engine are selected, there is only one pf at which the capabilities match. Somehow it has been determined that selecting 0.8 as the standard rating satisfies most customers in terms of buying neither an engine nor a generator that is larger than required.