The question deals with three scopes of government - the UK, the USA and internationally. I'll take each in turn.
The UK
Pottermore has a page on the UK Minister of Magic:
The Minister for Magic is democratically elected, although there have been times of crisis in which the post has simply been offered to an individual without a public vote (Albus Dumbledore was made such an offer, and turned it down repeatedly). There is no fixed limit to a Minister's term of office, but he or she is obliged to hold regular elections at a maximum interval of seven years.
The Muggle Prime Minister has no part in appointing the Minister for Magic, whose election is a matter only for the magical community themselves. All matters relating to the magical community in Britain are managed solely by the Minister for Magic, and he has sole jurisdiction over his Ministry.
(Pottermore, "Ministers for Magic").
From this we can gather that:
- The Minister of Magic is appointed through democratic means and that elections are held every seven years (although they can be more frequent).
- The Minister has "sole jurisdiction" over all magical governance in the UK. It therefore follows that they have the definitive say over decisions taken on matters of law-creation, law-enforcement, international statesmanship etc.
As to why no election is depicted in the books, Cornelius Fudge was appointed in 1990, one year before the events of the first book in 1991. We see two Ministers appointed in subsequent books, although these appointments happen in unconventional circumstances. Scrimgeour was appointed because Fudge was kicked out in disgrace. Thicknesse was appointed because Scrimgeour was overthrown. There may have been elections held for these appointments but it doesn't seem likely.
"My dear Prime Minister, you can't honestly think I'm still Minister for Magic after all this? I was sacked three days ago! The whole wizarding community has been screaming for my resignation for a fortnight."
(Half-Blood Prince, Chapter 1, The Other Minister).
"The coup has been smooth and virtually silent," said Lupin. "The official version of Scrimgeour's murder is that he resigned; he has been replaced by Puis Thicknesse, who is under the Imperius Curse."
(Deathly Hallows, Chapter 11, The Bribe).
So Ministers can resign but they can also be sacked, although who by isn't stated. Most likely, the Wizengamot have executive power to remove a Minister from power as they are the only other official legislative body shown in the series. They can probably also appoint a Minister under emergency conditions such as a state of war. Scrimgeour and Thicknesse appear to have been such appointments.
As for how the UK Ministry compares with its Muggle counterparts, the Minister seems to be much more powerful than the Muggle Prime Minister. The PM is constrained by the Houses of Parliament, for which there seems to be no magical equivalent. They also run their government through the Cabinet and through different government departments. There is a similarity with the Ministry of Magic here. There are different magical departments, each with a department head (e.g. The Head of the Department of Magical Law Enforcement) that operates like a Muggle cabinet minister.
The USA
Witches and wizards in the USA operated for many years without any effective government of their own.
As the wizarding community in America was small, scattered and secretive, it had as yet no law enforcement mechanism of its own.
(Pottermore, "History of Magic in North America, Seventeenth Century and Beyond").
When the Magical Congress of USA was created in 1693, it functioned as a de facto government.
Perhaps the most significant effect of Salem was the creation of the Magical Congress of the United States of America in 1693, pre-dating the No-Maj version by around a century. Known to all American witches and wizards by the abbreviation MACUSA (commonly pronounced as: Mah – cooz – ah), it was the first time that the North American wizarding community came together to create laws for themselves, effectively establishing a magical-world-within-a-No-Maj-world such as existed in most other countries.
(Pottermore, "History of Magic in North America, Seventeenth Century and Beyond").
The MACUSA is headed by a President. They head up the Congress - unlike their Muggle equivalent, who runs the White House and is distinct from Congress.
The International Scene
The various Ministries of Magic work together internationally under the umbrella of the International Confederation of Wizards. This is the magical equivalent of the United Nations.
The Chairman of the ICW is termed Supreme Mugwump. The appointment seems to be quasi-democratic since Dumbledore was ejected as Supreme Mugwump by a vote.
"They're trying to discredit him," said Lupin. "Didn't you see the Daily Prophet last week? They reported that he'd been voted out of the Chairmanship of the International Confederation of Wizards because he's getting old and losing his grip, but it's not true; he was voted out by Ministry wizards after he made a speech announcing Voldemort's return."
(Order of the Phoenix, Chapter 5, The Order of the Phoenix).
Since it was "[UK] Ministry wizards" that voted Dumbledore out I'd suggest that the Supreme Mugwamp must be nominated by their domestic Ministry and that that Ministry can subsequently hold a vote of no confidence in their original candidate. That would explain how Dumbledore could be voted out by just one Ministry out of hundreds.
Liechtenstein's Ministry of Magic originally refused to join.
He looked ahead for a question he could definitely answer and his eyes alighted upon number ten. Describe the circumstances that lead to the formation of the International Confederation of Wizards and explain why the warlocks of Liechtenstein refused to join.
(Order of the Phoenix, Chapter 31, O.W.L.s).
This suggests that membership of the ICW is optional and the Supreme Mugwamp doesn't have any real power of their own (much like the UN Secretary General). Their job is to bring the various Ministries of different countries together and secure international agreement where possible. Domestic governments could also interact with other Ministries themselves. For instance, the British, French Bulgarian Ministries worked together to put on the Triwizard Tournament, coordinated in the UK by the Department of International Magical Cooperation under Barty Crouch Senior.
See also "How Is the Minister For Magic Chosen and What Qualifications Are Required To Hold Office?".
Bonus point: the meeting referenced in the question was the Warlocks' Convention of 1709.
"But it's against our laws," said Ron. "Dragon-breeding was outlawed by the Warlocks' Convention of 1709, everyone knows that. It's hard to stop Muggles noticing us if we're keeping dragons in the back garden..."
(Philosopher's Stone, Chapter 14, Norbert the Norwegian Ridgeback).
It's not clarified what this convention was. It seems to be an international summit to ban the breeding of dragons, though, not a committee of the UK Ministry.