Interesting question. I haven't heard of any official efforts by the French or the British to promote secession but that doesn't mean there weren't any.
Not likely. British foreign policy during the ACW was to sell and trade with both sides to the greatest extent possible. That includes military sales as well. Yes the British did stop the sale of the Laird Rams when threatened with war by Secretary of State Stewart. The French sold less military hardware to both sides but would not sell the ship known as the Stonewall Jackson directly to the Confederacy but did sell it to a private concern in the Netherlands and then the Confedracy bought from the concern.
It's been over one hundred and fifty years since the ACW and if there is evidence of foreign intervention it should have come up.
Leftyhunter
Britain and Canada more or less "looked the other way" regarding CS operations. The CS Alabama, for example was constructed in Britain for the CS navy. The Confederate Secret service ran operations out of Canada against the US. Canada did arrest the St Albans raiders in late 1864, and returned the recovered money from the bank robberies. The raiders were later released.
I doubt the British would have wanted to provoke the United States - the fear that we might want to invade Canada (not yet an independent country) still lingered.