Did they have any right to hold such a referendum? If they do, how come the Basques don't?
As far as I'm concerned the Basque should have the right to hold a referendum.
I don't think the concept of a de jure president is coherent - the only way a constitution got written in the first place is when the de facto rulers chose so.
In what sense do you consider the Kiev government the de facto rulers of the whole of Ukraine including Crimea? The government got control over Kiev through the thread of using violence. It might have also got control of Crimea through the threat of violence if Russia wouldn't have protected the Crimean government from being violently coerced to follow the dictates of Kiev.
I don't see that when I president loses control over the capital because of threats of violence but that president still controls other parts of the country he automatically stops being president.
Protecting a minority in a country from being violently coerced is what responsibility to protect is about.
Whose law?
The new government in Kiev didn't change the part of the constitution that declares that you need an impeachment process to impeach a president and they didn't go through the process of impeaching the president that's outlined in the Ukrainian constitution.
Can the government of Taiwan authorize an invasion of China?
The Taiwanese government can certainly allow Chinese soldiers to do whatever they want within the borders of Taiwan.
Can Franz, Duke of Bavaria authorize an invasion of the UK?
The ruler of Bavaria had never authority over the UK and according to the German constitution he doesn't have any authority over Bavaria either. The president of the Ukraine on the other hand has authority over the territory of the Ukraine and at the time of the Crimea referendum he wasn't impeached via the the legal process of impeachment that's described in the Ukrainian constitution.
If you treating a coup on the capital on a country and the president leaves the capital I don't think that means that the president "resigned" or loses his claim on being the president.
A more fitting comparison would be if the Dalai Lama can authorize military actions of foreign troops within the territory of Tibet. But the Dalai Lama doesn't control any of the territory in Tibet while the Ukrainian president still had some backing in parts of the country, so the Dalai Lama has a weaker claim.
Let's say you would have a few US military generals who threaten a coup if Obama doesn't resign. Then Obama leaves Washington and a majority of the US congress makes a vote that John Boehner is the new president. Do you think that would mean that Obama loses his claim to being the president of the US?
The Taiwanese government can certainly allow Chinese soldiers to do whatever they want within the borders of Taiwan.
There are no de jure borders of Taiwan. Both the PRC (which currently governs mainland China) and the RoC (which currently governs Taiwan) (officially) claim that both mainland China and Taiwan are part of one nation and each (officially) claims to be the sole legitimate government of the whole nation. Also, the RoC used to control mainland China and was expelled from it in a way (IIUC) not completely unlike the former Ukrainian government or Obama in your hypothetical.
Some of the comments on the link by James_Miller exactly six months ago provided very specific estimates of how the events might turn out:
James_Miller:
Me:
"Russians intervening militarily" could be anything from posturing to weapon shipments to a surgical strike to a Czechoslovakia-style tank-roll or Afghanistan invasion. My guess that the odds of the latter is below 5%.
A bet between James_Miller and solipsist:
I will bet you $20 U.S. (mine) vs $100 (yours) that Russian tanks will be involved in combat in the Ukraine within 60 days. So in 60 days I will pay you $20 if I lose the bet, but you pay me $100 if I win.
While it is hard to do any meaningful calibration based on a single event, there must be lessons to learn from it. Given that Russian armored columns are said to capture key Ukrainian towns today, the first part of James_Miller's prediction has come true, even if it took 3 times longer than he estimated.
Note that even the most pessimistic person in that conversation (James) was probably too optimistic. My estimate of 5% appears way too low in retrospect, and I would probably bump it to 50% for a similar event in the future.
Now, given that the first prediction came true, how would one reevaluate the odds of the two further escalations he listed? I still feel that there is no way there will be a "conventional battle" between Russia and NATO, but having just been proven wrong makes me doubt my assumptions. If anything, maybe I should give more weight to what James_Miller (or at least Dan Carlin) has to say on the issue. And if I had any skin in the game, I would probably be even more cautious.