Lots of questions so I will split your question as follows;

Quote!which has flowered wonderfully this year (18 to 20 huge flower spikes).

Reply; This is because your plant is happy in its present spot!

Quote; The trouble is the stems droop and I have to stake them individually.

Reply; Did you feed them?

If you feed with a high potash feed from when the flower stem emerges this will siffen the stems!

Quote;Now they have all flopped and I wondered if I should lift and divide the plant.

Reply; Definitely not ( unless it is absolutely necessary!)

Agapanthus do not like root disturbance and if it were in a pot it prefers to be pot bound!

Which makes me think you have a fine rootball, and it is this that is giving you your success!

Quote; It still has all its stems and the leaves are still green and vibrant. Would it hurt the plant to cut back all foliage and stems.

Reply; What's the hurry? leave them to feed the root system for next year!

Let the 'frosts' dictate when you need to do this!

Then you will need to cover the rootball with a thick organic mulch to protect it throughout the winter months!

Quote; so that it can be dug up now in Autumn, or should I leave it until spring when it has all died back - but is this too late to lift and divide.

I'd appreciate your advice because it's a beauty and I'd hate to lose it.

Reply; I think my previous comments answer this!

Quote;Many thanks.

Your welcome! ...Tg

There is a bit more info here;

http://www.thegardenersalmanac.co.uk/Data/Agapanthus/Agapanthus.htm

Agapanthus love to be pot bound so leave well alone. if you have to move it then spring would be best as Teegee says.

Thank you both so much for your help. Your advice Teegee has put my mind at rest, I really didn't want to have to dig it up, and no, I don't feed it very often at all, so next year I will make sure I do just that, high potash - does that mean tomato food, or seaweed? Thanks again.

That makes me jealous to think you have an Agapanthus with 25 stems on the plant......wonderful. I recently found the remains of a clump I planted ages ago in a crowded border. It has never bloomed.....or I would have realised it was still there, mixed up with big Alliums. At the moment there are 5 parted roots in pots, to see how they go. All the advice by Teegee is very useful.