(1). When I cull my agapanthus do I keep the biggest ones that presumably have already flowered or the smaller ones that presumably did not flower. (2). When I dig them out how long can I keep them before transplanting and how do I keep them "alive".
There should not be a need to dispose of or cull your Agapanthus, except in the case of root rot or possible disease.
It is best to dig up, divide and replant the perennials as soon as possible.
If you need to wait, you can plant them into pots until the ground is ready.
Here are some links with more information.