1) Even if failing in the Atlantic seaboard colonies, the Patriot leaders could/would have set up a regime to the west of the mountains.
Sobel's "For Want of a Nail" might be responsible for this one, with its feature of refugee founding fathers creating a new state in Mexico.
In any case, not too likely at all the Patriots would succeed west of the mountains if they failed to the east. Despite the existence of settlements west of the mountains in places like Kentucky, the demographic center of the colonies was definitely the east. Hardly any majority or critical mass of Amerindians had reason to work with the American rebels, there was far more self-interest in working with the British. Among whites west of the mountains there was not a Patriot consensus either, there were many loyalists in the west as well.
2) In a perpetuated union of the 13 Colonies under the British Empire, the imperial capital is likely to move to America eventually, and not only to America, but somewhere in the middle of the continent rather than east coast
I've seen this a few times, not sure entirely where it comes from.
Countries are resistant to moving their capitals out of both tradition and self-interest. Decades after a failed revolution, the British Isles would still have a larger population and higher GDP than British North America.
Yes the circumstances of the Napoleonic war and the particular circumstances of the Portuguese empire result in the dynasty fleeing from Lisbon to Sao Paulo, but it's the exception that proves the rule. This did not happen in the Spanish Empire despite some advocacy for it in Spanish America. Additionally, there's not many likely scenarios in which the King and Parliament would be driven out of the British Isles.
Also, look at the United States. For all its westward expansion, and the major growth of the demographic weight of California and Texas (and the lack of corresponding eastern expansion), the capital has remained in Washington, DC near the east coast since the `18th century.
3) After defeating the Patriots, the reunified British-American Empire would most likely go on a conquest binge against Spanish America.
That would probably be expensive and uncertain of success. The prize would be alot of thinly populated territory considered desert, but the tropical areas of greater value would be disease-ridden death traps for the British, and they had experienced some of this before.
Additionally, the timing of the American Revolution after the great conquests of the French and Indian War was an object lesson in expansion not necessarily bringing stability
4) Every leader on the Patriot side would hang
This could happen depending on the particular circumstances of a British victory (like if it's overwhelming), but many of the PoDs for a failed revolution also involve diminution or elimination of French support. While that makes Yorktown impossible, to settle the bleeding ulcer, Britain is likely to have to end the war in a negotiated manner, the first thing the rebel side being sure to negotiate being their lives and property. Once agreed to as a package deal, there would be high costs in Britain politically in doing a bait and switch, conducting a star chamber and hanging the former rebel leaders.
5) Defeating the American Revolution would lead to a relatively stronger British monarchy over the long-term and a weakened parliamentary position
Sure there's one less example of a republic, but internal factors limiting the power of the King had been around a long time, and over the long-term pressures for broadening the franchise would continue
6) With no American revolution or a failed American revolution, there cannot be a French Republic.
Yes America was the *largest* republic of its day, but there were examples of republics from Europe including the admittedly short-lived Commonwealth of Britain, Switzerland, Venice and the United Provinces of the Netherlands.
If France gets involved in the American Revolution and loses, its finances are still screwed.
If France stays uninvolved but does not otherwise reform its fiscal systems, an Estates General is probably going to happen and be an opportunity for an explosion.
Some say that while an overthrow of the absolutist regime as it was was likely, but that the National Assembly would find a Republican government inconceivable without the American example.
But the lack of an American example still would not take away the ever-worsening alienation of the King and National Assembly. If the Revolutionaries decide they have to kill him, they are sort of a Republic by default. How would a new King be selected?