Harry was never an official Horcrux, according to J.K. Rowling. And, yes, once the fragment of Voldemort's soul that Harry contained was destroyed by Voldemort, Harry lost the ability to speak Parseltongue. I'm not sure what other powers you're referring to, but JKR has stated that Harry did lose the ability to speak and understand Parseltongue.
Nigel: Can Harry speak Parseltongue when he is no longer a Horcrux?
J.K. Rowling: No, he loses the ability, and is very glad to do so.
(J.K. Rowling Interview, The Deathly Hallows Web Chat,July 2007)
ETA: 3.28.14 In Chamber of Secrets, Dumbledore and Harry converse about Voldemort transferring some of his powers to Harry the night he tried to kill Harry:
‘Unless I’m much mistaken, he transferred some of his own powers to you the night he gave you that scar. Not something he intended to do, I’m sure ...’
‘Voldemort put a bit of himself in me?’ Harry said, thunderstruck.
Chamber of Secrets - page 245 - Bloomsbury - chapter eighteen, Dobby's Reward
As noted in the question, the "powers" transferred from Voldemort to Harry include Parseltongue, resourcefulness, determination, and a certain disregard for rules. Parseltongue, though, is the only item mentioned at this point in the story that is a bona fide power; the other three are character traits.
The only thing I can think of off the top of my head that qualifies as another power of sorts would be the connection between Harry and Voldemort's minds, which allowed them to, at times, read and/or "see" the other's thoughts. This is related to Harry's scar -- the more frequently Voldemort and Harry's minds connected, the more pain Harry would feel in his scar. In the epilogue in Deathly Hallows, it mentions Harry's scar didn't pain him after Voldemort's death. If you want to call their mind connection a power (of any level or sort), then it was lost once Voldemort died.