My area of knowledge is limited to corporate jets, so I’ll try to answer this 
The biggest single issue is FOD being ingested into engines. This can be caused by thrust reversers, intake forces, or nose wheel spray.
The Citation XL (which I teach on at FSI) USED to have a **supplement **(in fact is was #32) to the AFM (aircraft flight manual) that allowed for gravel runway operations. In ALL the jets I’ve flown, it was an operational limitation in the AFM to operate from paved surfaces only. It was EXPENSIVE to obtain the required supplement and a/c modifications that went along with it. The biggest modification required was to the nose gear. A small bleed air line was brought from the engine to the nose wheel. The nose wheel was retrofitted with a small “hamster cage” that would spin the nose wheel up to speed (using that bleed air) PRIOR to touching down. This limited the spray from the rocks that would otherwise occur when the nose wheel touched down and initially skidded. The supplement in question was used by one operator, and he was in Africa. It was never used by a US operator. So unless he came here, you would NEVER see an XL at a gravel strip. The supplement was removed from the AFM all together about 3 years ago, and as far as I know isn’t even available any more.
There were a few more limitations also such as the size of the stone used, runway shape, and compactness. The effect to landing distances was about an 70% increase. For take off it was about a 85% increase. For the example given:
What corporate aircraft (jets) can land at grass or dirt strips and take off (4000ft strip Sea Level)? At 4000ft of dirt, is the strip mostly limited to Citations and Learjets? Inside of corporate jets, is their a chart or performance plate for grass/dirt strips?
If you increase the strip length to 5000ft, does that 1000ft of dirt make a big impact? With 5000ft of dirt, could you land and take off in a Falcon 900 for example? Will NetJets, Citation Shares, etc fly into grass and dirt strips? Lastly, at 3000ft elevation, does a 5000ft strip of dirt/grass further decrease performance of a jet compared to 5000ft of pavement given the same weather conditions?
@ 3000’ on a standard day @ max t/o weight (18700, which is limited from 20000 b/c of the gravel) the XL would need 4260’, to land it would need 4240’. The same a/c being operated from a paved runway needs 3660’ 3070’ respectively.
I don’t have the FOM (flight operations manual) for NJA in front of me but I’m about 99% sure they are not allowed to operate from “unapproved” airfields.
No they’re not, just like the XL they have a supplement that allows for it, but the operator most posses that supplement. And generally speaking supplements are for individual a/c (ie N numbers) and not for entire fleets (ie all US Airways 737-200ER’s)
Again this is for FAA operated a/c, different governments might allow such operation with out additional manufacture in put.