The UN does want to be charge of international law, Mr. Mom. Who else can be? The tsunami relief thing was a mess, but there were a lot of NGOs who were accomplishing nothing and a lot of services being duplicated while other services were going undone, not to mention the political problems that the huge influx of aid workers caused.
The UN was in charge of organising the effort and were trying to get a handle on things. Should they have been forcing people to wear UN uniforms? I don't know, but they were trying to do what they were charged to do.
Let's have a look at international laws though.
Who writes them? The international community through the UN. Nations that are not signatory to various treaties do not have to abide by those treaties.
Who enforces them? The international community through the Security Council.
Who runs the international courts? Experts on international law from all over the world.
Who goes to international court? Countries have the option of trying their own war criminals etc. first. If they cannot or will not, then the Security Council can pass a resolution that those who have committed war crimes and crimes against humanity to try them. A country without the resources can request that the ICC hold a trial for them.
The UN has no money of its own, no police force, no military. Membership is not mandatory and it cannot levy taxes. It is not government in the way you mean.