The correct sentences are:
"The Republican Party began as a series of anti-slavery political meetings held in the Midwest in 1854".
The main reason for these first meetings was to establish an opposition to the Kansas–Nebraska Act which was trying to reinstall slavery in those states.
"During the election of 1824, Andrew Jackson won the most electoral votes, but not the majority needed to be elected President".
Of the four biggest candidates, none got majority in the Electoral College. So, according to the Constitution, the House of Representatives had to choose the winner and John Quincy Adams was elected.
"Third parties play an important role in the strengthening of the two major parties".
Third parties usually emphasize specific issues that are being looked over, putting pressure on the position of the major political parties to address these issues.
Hope this helps!