1) I have to say that, a coup against Lincoln is not going to happen since the tradition of democracy (even stronger than in the South) and civil control over military had been firmly established in the US. As for assassination, the US shrugged off (successful) assassinations multiple times historically. In the long run, US politics without the Southern bloc is going to be more stable, not less.
2) Prussia already surpassed Austria by 1866, in terms of industrial capacity and experience much stronger growth. Plus, the gap between Austria and Prussia was not as huge as Union versus Confederacy. The Union-Confederacy gap is closer to WW2 US-Japan gap than Austria-Prussia. Look, let me remind you that, the entire Confederacy had a smaller industrial base than *Massachusetts* in 1860 - this is the huge gap that I am talking about.
3) British and French support for the Confederacy would not prevent the US from growing more powerful over time. Plus, French support would totally depend Napoleon III’s position in France - forget about it if France becomes a republic led by the (pro-Union) Republicans. Alternatively, the CSA should forget about French support if it invaded Mexico under Maximilian.
A democracy? Maybe, but an authoritarian one more likely. I would view the USA as becoming more like a less extreme Apartheid South Africa.
You're right, the Confederacy would lose French support if it attacked the Second Mexican Empire. More likely they would just beat the shit out of Spain just to gain access to Cuba. Nobody cared much about Spain in TL-191 anyway. And turn Nicaragua into a protectorate for good measure so that they can build their Nicaragua Canal. Maybe Richmond could even get into a scuffle with the Japanese over the Sandwich Islands (provided the British have their hands full in Canada to ward off a more aggressive USA) and spark a Confederate-Japanese Great Game in the Pacific.
Or alternatively, as opposed to the USA weakening, when the CSA ends up buying Sonora and Chihuahua from the Mexicans with the blessing of the French (this actually happened in canon TL-191), the Mexicans of Baja California, resenting Mexican weakness, form a breakaway Restored Republic of Mexico in Baja California, only for the USA to intervene and annex Baja California shortly after (the Confederacy didn't want to test the limits of French-CSA friendship so they remained neutral), drawing plenty of ire from the international community. Ironically enough, the USA doesn't weaken so much as it becomes more and more like the Confederacy, what with many in the Union still blaming the black population for the disastrous American Civil War, and says, "Eh, f*** it" and allies with the CSA because the Confederacy was one of the few nations that recognised the Union's control over Baja California (the CSA wanted the North to recognise Sonora and Chihuahua as legitimate Confederate territory in exchange for recognising Baja California as legitimate Union territory). Moreover, both sides would rather not fight each other with the growing Japanese threat in the neighbouring Kingdom of Hawaii (which is now a Japanese protectorate), especially considering Japan's opportunistic annexation of the Galapagos while a revanchist Second Republic of Gran Colombia was busy re-conquering Ecuador. That, coupled with the French gradually reducing their commitments to Mexico thanks to mounting tensions with an increasingly powerful Kingdom of Italy right next door, would likely cause revanchism to flourish in Mexico and the Second Mexican Empire could turn to another power, like say the Empire of Brazil (the Brazilian monarchy was never abolished ITTL) or the Japanese Empire, as both would be increasingly at odds with the CSA due to Richmond becoming a bit too big for its britches.
In TL-191, the British and French didn't just give token support to the CSA for the war in 1882; they attacked the US directly via Canada and Mexico. British troops even raided the San Francisco Mint for goodness' sake. Who is to say they wouldn't do so here provided the CSA doesn't burn its bridges with them just like TL-191?
Oh, and
here is the inspiration for my idea. One key difference is that it is not going to be Britain + France that goes communist ITTL. I'm kinda leaning towards Brazil or Japan for TTL's USSR.