1) Yes; 2) Quite possibly; 3) No
In OTL, William Seward pursued quiet diplomacy in regards to kicking the French out of Mexico. However, there were many politicians who argued for saber-rattling and military aggressiveness. President Andrew Johnson himself wanted a more aggressive approach. Assume William Seward dies somehow. Also assume that his replacement is one of the saber-rattlers. Do the French still give up when they did? Do they do it sooner? Later? In a worst-case scenario, could France and America go to war?
1) Yes; 2) Quite possibly; 3) No 4) Also, no...
First off, the French/Mexican Conservatives had lost the war against the Mexican Liberals (in a geostrategic sense) the day Lee surrendered at Appomatox. The fact that the US was able to dispatch an army of 50,000 - under Phil Sheridan - to Texas in 1865 made that undeniably clear.
Second, the Liberals, given American supplies and the ability to recruit American specialists (field artillerymen, notably) were able to defeat the French/Conservatives in the field on their own; pretty clear by the failure of the French offensive in the winter of 1865-66, when the French withdrew from Monterey, Saltillo, and Tampico.
Third, when the Prussians beat the Austrians in 1866, even NIII understood his adventure was over because France had other issues, much closer to home.
Fourth, by the spring of '67, the French only controlled Mexico City, Puebla, Queretaro, and Vera Cruz, plus the corridors connecting them; not much. Carlotta was insance, Maximillian was ready to abdicate, NIII had ordered Bazaine to withdraw, and in fact had ordered Bazaine to negotiate with any of the liberals but Juarez - no takers.
Fifth, Marquez and Miramon showed up and persuaded Max to keep fighting; the French withdraw after negotiating safe passage with Diaz in Feb.-March, 1867.
Max et al have 10,000 at Queretaro; the Liberals have some 40,000, including units equipped with Spencers, and US-manned artillery; not surprisingly Max et al surrender in May, with the expected results...
So, no. The French intervention only occurred because the US was in the middle of the Civil War and the Mexicans were exhausted because of the Reform War; as it was, First Puebla was a pretty clear indication of what the French could expect, and yet they still blundered along for another four and a half years, squandering resources they could ill afford to lose given the pending confrontation over what nation state was going to control Central Europe.
Best,