Regarding the questions referring to the responsibility of the SAR costs, if my rusty memory is correct it was all set out in the original 1944 Chicago Convention (the one that established ICAO) or perhaps some later amendment. Adhering States provide Air Traffic and SAR services to all flights falling under the Convention, and in return they may collect an Air Navigation Charge proportional to the overflown distance modulated by an aircraft weight formula to recover these costs. While purely domestic flights do not fall under the Chicago Convention, most adhering States apply the same rules to domestic traffic too. Same Convention sets out responsibilities for accident investigation, and any related search and recovery costs are bourn by the investigating authority (in principle also funded from the Air Navigation Charge). The picture becomes complicated when a small country (with proportionately small ANC income) needs to fund a major investigation (usually involving a foreign airline), but in this case they have the option to delegate the investigation to another interested party (State of operator or manufacturer).
Of course in real life in most countries the collected charges just get swallowed up by the larger state budget, and there is some moaning involved when the bill is presented for an expensive recovery operation (usually carried out by specialised salvage firms).