Generally speaking in international law, invading another country without explicit UN approval is illegal unless that country has attacked you first.
Not quite. International law & UN charter are two different things. There are two cases where the UN charter allows force - self-defense, and after UN authorization of force (see chap VII, sect 39). But self defense may extend to allies, and may involve first strike in case of potential imminent attack (see 'Caroline Test'). International law on the matter generally aligned with this concept, is far more difficult to describe accurately.
In the case of the US&UK vs Iraq 2003, the US&UK argue in two ways. That Iraq's refusal to abide by their agreement to permit unrestricted UN inspections was a direct threat to the US & allies. Also that since Iraq violated it's 1st gulf war cease-fire agreements, that this was a continuation of the 1st war, where the 1998 UN resolution permitting force was the controlling document.
Note that international law (not UN charter) recognize belligerence (such as shooting at no-fly jets) as an act suspending peace treaties. For example it was determined that Nazi German belligerence in the 1940s suspended the 1919 Versailles Treaty.
As a practical matter, Israel didn't (and couldn't) ask for permission for the 7 day war, US didn't ask permission to invade Angola nor bomb Kaddafi in the 1980s, Russia didn't ask permission to attack Chechins or Georgia. Iran/Iraq didn't get authorization for their war. Iraq didn't ask permission to invade Kuwait. So these UN charter rules wrt war are more often exercised by violation than adherence.
If Saddam had been involved in 9-11 I'd say it qualifies as a legitimate reason to attack Iraq.
Certainly, but that was
never the Administration's case.
By violating a 1991 cease-fire agreement (ejecting inspectors, violating the no-fly zone), Iraq created a resumption of hostilities. The US&UK argue the preceding authorization of force persisted. More arguable - that the failure to abide by inspection agreements meant to find/destroy WMD made Iraq an imminent threat to US&Allies.
I think the US&UK are safe wrt to international law and courts, but are on the very edge wrt UN charter violations. Other nations have been far over that edge w/o actions being take.
In practical terms this doesn't limit any of the permanent members of the US security counsel because they get a veto over any declaration an invasion is illegal.
The UNSC is not a court, but they oddly do condemn acts and make claims that some violate international law; w/o seeking prosecution. But Kofi Annan could bring a case at the International Court of Justice or International criminal Court (neither is UN) claiming the US was guilty and the UNSC has no veto power there. He could not eject the US or UK from the UN since that requires a UNSC vote. He could I believe suspend US & UK general assembly voting privileges (not on the SC).
An actively in progress genocide would be not only a valid reason but would demand action. this is why many countries, including the US, were reluctant to acknowledge Rwanda as a genocide until after the fact. Doing so would have required them to send troops to help stop the genocide. If the US had attacked while Saddam was actively slaughtering Kurds it would have met this test IMO
No. The UN in 1948 passed a Convention against Genocide (Resolution 260 (III)) which then becomes part of international law. However the UN charter specifically states (chap I, art 2 , pp 7)
Nothing contained in the present Charter shall authorize the United Nations to intervene in matters which are essentially within the domestic jurisdiction of any state or shall require the Members to submit such matters to settlement under the present Charter; but this principle shall not prejudice the application of enforcement measures under Chapter Vll.
So the genocide has to threaten international peace (chap VII) before the UNSC can act w/ force.
In Rwanda, w/ the governments agreement, the UN sent observers and later, after a peace treaty was signed, a small "assistance mission" to help parties implement peace. Then after some major setbacks it decreased the peacekeepers and created an arms embargo. Only then with violence spilling over borders did the UNSC authorize creation of a force under chap VII, strictly for humanitarian purposes wrt refugees.
Serbia/Kosovo/Yugoslavia resolution also included multinational peace and refugees before force was authorized.
So in the unlikely event you can keep genocide within your own borders - then the UN will issue feckless statements and do little.
The problem, I think, is that US Congress declaring genocide compels action politically and morally - not via UN or law.