Was not a large part of New Orleans' draw though that it would serve as a short cut to the west rather than having to take the standard long route from New York, Boston and the like.
I wouldn't see New Orleans being such a draw here.
No, its main draw at first was very simple: more boats left Liverpool for New Orleans than anywhere else in North America. And while many did eventually move upriver to St. Louis and Chicago, back then New Orleans was the third largest city in the US, and OTL wouldn't fall out of the top 10 until 1900. After French/Creole, the main cultural influence to this day in N.O. is Irish. I myself am part Irish, and I live in a part of town called "the Irish Channel." As for other European groups, the part of town now called the Bywater was known as "Little Saxony" in the mid-1800s, and the French Quarter by 1900 was a thoroughly Italian neighbourhood. A good example of the cultural diversity here in the 19th century is the density of Catholic churches, each serving a different community. In my immediate area (8-10 block radius), we have Our Lady of Good Counsel (for the Irish), St. Henry's (Germans), and St. Stephen's (French).
Also, we're looking at immigration in the 19th century through 21st century eyes. For many of these poor, un- or barely-educated, often illiterate people, they didn't realize that there was more than one country in North America. Many of the ancestors of the large Italian and Jewish populations in Montréal, for example, weren't people choosing Canada over the US but instead ended up buying passage on a boat that wasn't landing at New York or Boston. Given the amount of traffic that passed and still passes through the port of New Orleans, it would have remained a major point of debarcation for Europeans well into the 20th century. Whether they stay or not is a good question, but there can be no question that European immigrants would have arrived in large, if not impressive numbers, into New Orleans and other Southern ports, since, if for no other reason, many of those arriving won't know the difference until they get there
As for Britain and cotton-
No, during the ACW other,better sources of cotton popped up. Britain was largely switching on these.
Then of course there's the moral factor; you'd get people embargoing companies that use confederate cotton forcing them onto more humane Egyptian stuff.
Actually, the British used Egyptian cotton (better quality, but much more expensive) as a stop-gap, but in late '62, as it became clear that a Southern victory would be long in coming (if ever), they began a process of switching from opium poppy to cotton production in Bengal. Partly this was because of the severe supply crunch, partly because of "humanitarian" concerns (it's hard to bring Christ to the heathen Chinese when they're always doped up), partly because the Chinese had finally been forced to accept payment for tea in other than silver (silver the Brits got by selling opium to the Chinese), and partly because the British had finally begun producing decent quality tea in Ceylon and Assam, creating less of a need for the Chinese variety.
In a best-case scenario for a CSA vistory, say an end to the CW by no later than late fall/early winter 1863, the Brits will be able to cash in their cotton bonds and will continue for some time to buy the majority of their cotton from the South, if for no other reason than it's cheaper to buy and ship from New Orleans, Charleston, Savannah or Wilmington. Suez wasn't built until 1869, and until 1882 it was under Franco-Egyptian-Ottoman control. Indian cotton wasn't a "better" source at the time, it was a "British" source. They'll still continue to develop cotton production in Bengal (for security reasons if nothing else), but probably at the same time try other crops in their switch from opium cultivation since the supply crunch will have been alleviated. Over time, the Union and France will replace Britain as the main buyers of cotton, since neither will have any other ready source for it, and other countries, such as Germany, will enter the market, and their most likely source will be the CSA.
During this time, the mid-late 19th c., the CSA will begin to transition to a more industrialised economy. The Civil War will have taught the South a good lesson, and many of its early leaders were quite progressive thinkers for their class, place and time. Besides, by the outbreak of the CW, slaves were already becoming too expensive to buy and support. It's horrible to talk about people in these terms, but a young, fit black field hand in good health in 1860 sold in New Orleans for up to $1000 (£220). This is unskilled labour I'm talking about!