Once an iris rhizome has been dug and replanted in a different location, it normally takes a of years to get established and produce to its full capability. It is nice that you did get a few blooms the first season.

A double blooming iris indicates that the blossom is a double rather than a single. blossom; the petals double up to cover the center and creates this rose like look to it.  I just want to be sure you meant "double-blooming" rather than "re-blooming."

The flowers usually last a few days but this depends on the weather and the variety. if the weather is mild and given plenty of water, blooms can last a little longer, but if it is hot and dry, it can stress the plant and kill off its blooms to focus on developing the root system. 

The mushy, gooey remains are normal for all varieties of iris blooms once they begin to die.  You should have a better outcome next year once the irises are established.

If your irises receive good rainfall, they generally do not need additional water once they are established.  For the first year, it's a little more important to keep the soil reasonably moist (not soggy) but well drained.  After that, just treat them like your other irises--double blooming or not.

A reblooming iris should put on another show later in the season, say late August or September.  Again, first year maybe not.