Why were proslavery southerners and the Pierce administration they controlled so eager to push for further American expansion into Nicaragua and Cuba and elsewhere in the 1850's?

Answer 1

Basically, pro-slavery Southerners like William Walker wanted to conquer lands like Cuba and Nicaragua so as to expand the territory that would be open to slavery. This idea was spelled out most famously in the Ostend Manifesto, which advocated taking Cuba by force if necessary. By this time (the 1850s) the issue of slavery seemed to be decided with the continental US. There seemed to be no room for slavery to grow. Therefore, many Southerners wanted to take more land outside the US so as to increase the power of the slave states relative to the free states.

With that said, even some politicians in the North also had plans for Cuba.

Answer 2

Because the aztecs discovered the cotton gin so they wanted to get in on the reaping of the benefits and profits of cotton planting/harvesting