I'm plotting an AU that has Harry transported to another world when he gets AK'd by Voldemort in Book 7. I haven't read that or any of the Potter books since basically the first week they came out and I've forgotten a lot. I don't really want to reread them in their entirety to get a detailed list of Harry's abilities so...

What wandless powers is Harry capable of at the end of Book 7?

The only ones that jump out to me are: Apparation and a very basic Occlumency.

If you've got anything more it'd be appreciated!

 

I don't think that Apparation was wandless, at least there are no examples of anyone apparating without a wand on them that I can think of. His Occlumency was actually very strong but it required him to be in the correct emotional state of extreme grief he was in after Sirius/Dobby's deaths. When he is able to defend his mind against Voldemort in person, the strongest known Legilimens in the series.

Apart from that I don't think there are any examples of wandless magic performed by Harry, outside of the accidental magic he did in books 1/3.

 

No.

It was confirmed that you had to at least have a wand on you to Apparate, though you don't have to be holding it. I don't remember the source, though I believe this to be a JKR interview.

 

I just reread the section where they learn to apparate and skimmed all the other apparitions in book 6 and there is virtually no mention of wands. Moreover when apparition is taught there is no mention of wands in the lesson.

I might end up not using apparation at all. In any event the harry potter wiki says that the answer is unclear due to the following: Apparently in Book 7 Harry questions whether someone can apparate without a wand. However in Book 3 Cornelius Fudge suggests that Sirius escaped via apparition without a wand. They posit it's viable to apparate without a wand but that it may be more difficult to do so.

 

I say you can apparate without a wand. It's not a spell you cast, and there's no incantation. I don't see how not holding a wand but having it in your pocket would make any difference. Many witches and wizards choose not to apparate anyways since it is very difficult, and the chance of splinching is always there.

Other than that, most of the stuff Harry has done without a wand is accidental magic. Book three he makes the cupboard door burst open after he got angry at Marge, but again, that's not something that is controllable.

 

Also the basic premise of the fic is that Harry is reincarnated in Lu Ten right before the events in Avatar the Last Airbender. It happens the day after Lu Ten was buried and Harry essentially wakes up in a coffin with only his memories and grievously injured. He breaks out and tries to find a life for himself but runs into various obstacles and injustices that cause him to search out magic. Basically this story would be a journey to both heal himself, find magic, and intervene when he can to prevent injustice due to his saving people thing. His main obstacles will be that he lacks a wand, possibly a hand, friends, or any knowledge of the world he's in. He'll overcome or find some of that by the end of the story. If it ends up working out I have tentative ideas to continue it with other arcs that may feature the main characters from Avatar: The Last Airbender. In the first book most of the Avatar characters that he encounters will be relatively minor.

 

This. Details like that are so minor that you can write it whichever way you want without even bothering to justify it. For example, Pottermore says that brewing a potion requires waving a wand, something that's (arguably) directly contradicted by Snape in his very first appearance.

And OP, deviations from canon are quite welcome. If you want to make Harry able to perform Summoning and Banishing Charms wandlessly (albeit very weakly at first), do it. Having him work on that would allow him to fit in with Avatar:TLA world quite nicely.