Group DMs are not yet available through the API. Its not clear if/when they will make Group DMs available via the API.
As for GroupTweet, only a small subset of our users use GroupTweet for Group DM type purposes. I'm optimistic that this native functionality will educate the masses on the possibilities of Twitter Group DM's which will lead "power users" to look to GroupTweet to provide a more "full featured" group DM experience.
Working on a blog post to illustrate the GroupTweet advantages, but things like:
1) A GroupTweet Group DM account is permanent compared to a more ephemeral Group DM thread. No need to figure out who to add to the conversation each time, etc.
2) Never lose your messages. Full archival capabilities & storage with a GroupTweet Group DM account. Twitter will only display your 100 most recent DM conversations.
3) No limit on number of group members vs Twitter's native Group DM limit of 20 people.
4) Full control over who is included the the Group DM. With Twitter's native functionality, any group member can invite anyone else into the conversation. With GroupTweet, the group admin(s) have full control over who is allowed in the group, who has posting privileges vs read-only, etc.
5) With a GroupTweet group DM account, you get all the advantages of a full fledged Twitter account. Things like Twitter analytics, GroupTweet contributor analytics (which members are participating & sending the best messages), downloading your Tweet history, linkable messages, etc.
6) More customizable notifications. Not all or nothing approach for regular DMs.
7) GroupTweet Group DM's will work with every Twitter client. If you try and start a Group DM with your friends that are using Tweetbot, Hootsuite, or any other third party twitter client, they won't receive the message.
The direct capture and upload has been sorely needed, should make on-the-ground reporting much easier.