Could I propagate the flower stalk? Should I wait? Which method should I use?

You can propagate from leaves off the flower stalk. Sometimes you can root an entire flower stalk but once it's done flowering, the show is usually over.

You need to get the entire leaf intact, all the way to the base. I like to use a paring knife to cut into the flower stalk behind the leaf so I get it all. Photos here of the process:

Once I collect the leaves, should I remove the flower stalk?

You may as well enjoy the flowers. The little bit of carving I do doesn't interfere with flowering at all.