There are two questions here, so this response will address the second, about the state constitutional conventions of 1867 and especially 1868. These conventions were significant because they marked the ascendance of (in most former Confederate states) Radical Republican governments. By this point, the Southern states were divided into military districts, and states were required to ratify the Fourteenth Amendment. State legislatures, dominated by Republicans, called for constitutional conventions. Generally speaking, the constitutions they ratified established legal equality for African Americans, guaranteed them the right to vote, set up systems of public education, and rolled out other sweeping reforms. The right to vote for African Americans was required by Congress and was later mandated at the federal level by the Fifteenth Amendment.
In short, the constitutional conventions marked a brief experiment with multiracial democracy in the wake of the Civil War. As Reconstruction collapsed in Southern states, many of the reforms implemented in these conventions were rolled back, and many states called new conventions beginning in 1877.
Since you asked two questions in this post, I will answer the first one. President Lincoln and the Radical Republicans differed on Reconstruction. President Lincoln wanted a Reconstruction Plan that wouldn’t be too hard on the South. His plan, known as the Ten Percent Plan, called for at least ten percent of the population of a state to pledge loyalty to the United States. These states would then develop constitutions that banned slavery. President Lincoln would offer amnesty to all white southerners. This provision didn’t apply to the former leaders of the Confederacy, though. President Lincoln also urged the states to give African-Americans the right to vote.
The Radical Republicans wanted a harsh plan of Reconstruction for the South. They wanted to give voting rights to the former slaves. They also wanted to prevent the former Confederate leaders from voting. They wanted to provide money for the construction of African-American schools. They also hoped to give some land that belonged to the plantation owners to the former slaves. The 14th and 15th Amendments were part of the Radical Republican plan of Reconstruction. They also divided the South into five military districts and established the process for states to write new constitutions.