What are the Babadook's powers/abilities?

To understand Babadook's powers/abilities, one must know, first of all, what exactly is the Babadook? Is it an evil spirit, a mythological monster, an actual creature or just a symbol?

Babadook is the impersonation of Grief & Problems.

The wikipedia entry itself answers a lot of questions

Writing for the Daily Beast, Tim Teeman contends that grief is the "real monster" in The Babadook, and that the film is "about the aftermath of death; how its remnants destroy long after the dead body has been buried or burned". Teeman writes that he was "gripped" by the "metaphorical imperative" of Kent's film, with the Babadook monster representing "the shape of grief: all-enveloping, shape-shifting, black".

Amelia is a single mother grieving over her husband's death and finding it very difficult to cope with the responsibilities of parenting.

The director Jeniffer Kent has stated that

she sought to tell a story about facing up to the darkness with ourselves, the "fear of going mad" and an exploration of parenting from a "real perspective". In regard to parenting, Kent further explained in October 2014: "Now, I’m not saying we all want to go and kill our kids, but a lot of women struggle. And it is a very taboo subject, to say that motherhood is anything but a perfect experience for women.

With this background, let's go back to your original question: What are the powers of the Babadook? This question can be rephrased as What are the negative-effects of not being able to cope up with your grief and your problems?

Babadook can do what grief and problems can do to someone, someone who is not coping well. And this very thing is very subjective and varies from person to person, situation to situation.

And how is the monster defeated? By acknowledging it. Acknowledging one's problems and standing up to them. I think at the end, Amelia acknowledges and accepts the fact the her husband has indeed gone forever and she must overcome the hardships of being a single mother. This eventually weakens the monster within her and she feels empowered again.

From the same source:

Teeman states that the film's ending "underscored the thrum of grief and loss at the movie’s heart", and concludes that it informs the audience that grief has its place and the best that humans can do is "marshal it"

As for Sam's magic tricks, I see it this way: What happens to kids who have troubled parents at home? They are troubled themselves, not able to do good themselves. And what happens when their parents become strong? They too become strong and able to unleash their unlimited potential. Hence the great magic tricks.

The word Babadook is a anagram of "a bad book". In the book was a passage about seeing what the babadook really was underneath would scare you to death. The book was her suppressed feelings toward her son and the death of her husband.