Hey there,
There is lots of good rivers to paddle in Iceland. Whether you are looking for waterfalls, creeks, big water etc. August is as good time as any for most of the rivers. Still there are some that only run in springtime/early summer when we have snowmelt.
Regarding equipment, there are no places that rent out boats and there aren“t any shops selling ww-boats so you will need to bring your own. I have only tried flying my boat from here with Icelandair and all was good. They have a fixed price for kayaks:
www.icelandair.co.uk/information/baggage...t-equiptments/large/
Here you find our river guide - both the old and the new version (the google map version):
kayakklubburinn.is/index.php?option=com_...iew&id=158&Itemid=89
Here you my YouTube channel with a few videos to give you an idea of our rivers:
www.youtube.com/user/joimeistari/videos?flow=grid&view=0
Hope this helps,
Cheers,
Johann