I believe the British were planning an expedition to New Orleans by the time the Treaty of Paris rolled around. I'll have to look for the link, though...
I believe that the British had actually already fought
the Battle of New Orleans after the Peace of Ghent was signed (I think you're getting confused with the Treaty of Paris 1783, which ended the ARW, and the Treaties of Paris 1814 and 1815, which ended the Napoleonic Wars), so yes
Fair play. My knowledge of the South Sea Company extends to that they caused the South Sea Bubble and financial collapse of 1720, and ceased to exist. Still, the only attempt the British ever made to capture the River Plate was 1806-7 as far as I am aware, so it can't have topped their priorities...
At this point in time Lousiana would have still been French - it was only given away just before the peace negotiations - but I agree, a likely target. What about Lima? Cartagena?
Yes, true, Louisiana was. I mentioned it as a kind of "across all eras" answer where in this specific situation, yes it wouldn't have been a Spanish territory.
An attack on Lima is probably unlikely unless the Spanish really were in collapse as the Spanish could resupply it far, far easier than the British could (it's really not easy to resupply a military campaign on the Pacific coast of the Americas by sea, especially when your enemy commands the crossing points).
An attack on Cartagena is more realistic, and in 1758 the British did blockade it causing a major sea battle, and in 1741 a British force laid siege to the city and tried to capture it, but on neither occasion was the city actually captured by the British.
With Cartagena you have to bear in mind that it is the kind of location where there are a lot of diseases and marshland conditions that European (read: British) people are not used to, so while Britain may want to capture it, they don't want to have to hold it - it's the kind of location you attempt to take to force an end to a war. Also, while the British would surely love to capture it, taking the Viceroyalty of New Granada would give the British a huge land border with the Spanish empire to defend, so - it kind of depends when exactly you are talking about. At some times, it would be a tempting target, at others you have to admit that the time isn't right and just use it as a barter point. It's a bit hard to call IMO.