Short answers, pulled from the story.
Ludovico III Gonzaga commissioned the church in 1472 on a site occupied by an older Benedictine monastery. Construction more or less ended in 1494 during the initial phase.
Leon Battista Alberti designed the exterior using the scheme of the ancient Arch of Trajan at Ancona. He integrated a lower order of fluted Corinthian columns with a giant order of taller unfluted pilasters to combine two distinct architectural scales into one unified design.
The massive dome took fifty years to build between 1732 and 1782. Filippo Juvarra designed this great dome that crowns the church today.
According to tradition Roman centurion Longinus scooped up earth containing the blood at the foot of the cross and brought it to Mantua where it has remained ever since. Pope Leo IX recognized this relic as authentic in 1053 after it was rediscovered around 1049 in the presence of Matilda of Tuscany.
One chapel houses the tomb of early Renaissance painter Andrea Mantegna alongside a bronze figure of him created by Gianmarco Cavalli. Mantegna's own Holy Family painting also resides in that same space.