To address the first part of your question, yes, Union soldiers were paid. One of the closest things you can find to a constant across world history is that you basically always have to pay the army.

To address the more complicated part of your question, the answer is: it depended.

Some soldiers did enlist in the Union army to end slavery, but by no means all. Many soldiers enlisted because they believed in the cause of the Union. They thought that the Union of American states was a unique and special thing, and had to be preserved at all costs. Many Union soldiers were motivated largely by this sense of unionist nationalism. People who held this kind of position often believed that southern secessionists had endangered liberty and republicanism itself by taking their states out of the Union for selfish reasons. It is very difficult to measure how many people enlisted for which reasons, as opinion polling the way we understand it now did not exist, but if the profile of Union soldiers at the beginning of the war closely tracked with the broader northern public, Unionism was a more powerful force than a desire to end slavery. Again, there were people who did explicitly join the Union army to end slavery, but it was not necessarily everyone's dominant motivation. There is even some evidence that there were Union soldiers who deserted after the Emancipation Proclamation; they had signed on to preserve the Union, not free slaves. As a motivation to join the Union army, ending slavery was a far more important motivator for black soldiers than it was for white soldiers.

Readings

Chandra Manning, What This Cruel War Was Over: Soldiers, Slavery, and the Civil War

James McPherson, For Cause and Comrades: Why Men Fought in the Civil War

Gary Gallagher, The Union War

Elizabeth Varon, Disunion!: The Coming of the American Civil War, 1789 - 1859