I am not an expert on mid 19th century France. Off hand going to war with the United States on a risk vs benefit reward would show France had more to loose then gain .
Not really, at least not short term. Historically what held back Napoleon III (never exactly a long term thinker!) from getting involved on at least one occasion was that the British indicated they would object.
The Union has essentially no way to harm France and France has a lot to gain, at least from Napoleon III's PoV. They get prestige, a freer hand in Mexico (which doesn't use any of the same resources as this intervention would if kept naval) and of course they also get cotton (which was a major thing, though not enough to provoke war by itself).
If the French also made a comparatively small landing, they could also totally turn the war around. FWIW I think the most devastating time for a French intervention to show up would be one of these two options:
1) Last week of June 1862. This would essentially turn the Seven Days into the "Surrender of the Army of the Potomac", as with French squadrons off Fort Monroe in force McClellan's army has no supply line whatsoever. Combine this with a naval squadron turning up off New Orleans and off NC, and you can end up forcing the surrender of (checks ORs)
14,772 Army of the Gulf
16,296 Middle department
15,782 Dept. of North Carolina
21,630 Dept. of the South
113,895 Army of the Potomac
About 180,000 (Aggregate Present) out of an army only 500,000 strong by the same measure (actually more because the Seven Days casualties haven't happened). It's a gutpunch for the Union.
2) Late 1864.
Grant and Sherman's army both relied on sea supply at this point, so you could force the surrender of both by the same measure. With both of them and Butler, that's over 260,000 AP out of 600,000 and is thus an even larger fraction.