+1 Slim's condolences.
Some general observations.
The real "owner" is probably the operator of the cemetery e.g. the local authority. I suspect that when they "sell" a grave initially the purchaser is actually buying a right to interment(s) in that plot.
Most cemeteries will probably have a set of rules or regulations regarding entitlements to rights of burial in any plot. I would expect that the issue of erection of headstones probably follows similar lines.
Cost is also a factor here especially where you are talking about the purchase, erection and inscribing of the initial headstone to be placed on a particular grave. Most cemeteries will require any such proposed headstone and or inscription to have their prior permission and they may have limits or restrictions on size or style. The stone mason usually handles that. I don't know the present cost of a new headstone but I would expect that you could be talking about a few thousand Euro.
You ask about "control of his paperwork". That would usually vest in the executor or the administrator of the estate. If your late father's wife (now as his widow) is still alive she acquires certain legal rights over matters but not necessarily over everything. That said, even if your late father's estate is entirely made over to his wife, that probably gives her or the estate NO RIGHTS over the grave if it was purchased or owned by your cousin who simply allowed your father's interment in his (cousin's) grave. If the grave belongs to your cousin the rights are effectively his as to what happens about the headstone.
If your cousin bought the grave on behalf of the next of kin - as their agent - that might be a different matter. However, you would need to see whose name is on the official paperwork issued by the cemetery as that is the entity that the cemetery proprietors will likely regard as the formal purchaser / owner of the grave.
P.S. I see the posts before this. It might well be that your father's estate might have to offer to buy the grave from your cousin to obtain effective control over it. There might be a technical hitch there are some cemeteries may not allow transfer of ownership of a grave ! It depends on what they are prepared to allow under their rules.