Hedwig was Harry's pet but she was not always kept in the cage like muggles (we) keep our pet birds. Other than in Harry Potter and the Chamber of Secrets, Hedwig was allowed to fly out to deliver letters and also allowed to hunt.
There are references in the book about this:
Harry copied these words on to three separate pieces of parchment the moment he reached the desk in his dark bedroom. He addressed the first to Sirius, the second to Ron and the third to Hermione. His owl, Hedwig, was off hunting; her cage stood empty on the desk.
Harry Potter and the Order of the Phoenix, Chapter 3, The Advance Guard
Hedwig hooted happily at Harry from her perch on top of a large wardrobe, then took off through the window; Harry knew she had been waiting to see him before going hunting.
Harry Potter and the Half-Blood Prince, Chapter 5, Excess of Phlegm
The owls at Hogwarts Owlery were free to move out of the school for hunting. There are couple of references to this
During stay at Hogwarts:
The sun was high in the sky now and when Harry entered the Owlery the glassless windows dazzled his eyes; thick silvery beams of sunlight crisscrossed the circular room in which hundreds of owls nestled on rafters, a little restless in the early-morning light, some clearly just returned from hunting.
Harry Potter and the Order of the Phoenix, Chapter 14, Percy and Padfoot
The many perches were half- empty; every now and then, another owl would swoop in through one of the windows, returning from its night’s hunting with a mouse in its beak.
Harry Potter and the Goblet of Fire, Chapter 29, The Dream
I am giving these examples because even though Hedwig was Harry's pet she almost lived her life like a wild owl. Going out at night, hunting and stuff. Normally birds/animals that do hunting do not need special attention (as long as they are not injured).
Talons are made of keratin and continue growing throughout a bird's life as they are continually worn down through use.
Referred from http://birding.about.com/od/Bird-Glossary-S-T/g/Talon.htm
Harry did care about Hedwig.
Hmm,” said Professor Grubbly-Plank, her pipe waggling slightly as she talked. “Looks like something’s attacked her. Can’t think what would have done it, though. Thestrals will sometimes go for birds, of course, but Hagrid’s got the Hogwarts Thestrals well-trained not to touch owls.”
Harry neither knew nor cared what Thestrals were; he just wanted to know that Hedwig was going to be all right. Professor McGonagall, however, looked sharply at Harry and said, “Do you know how far this owl’s traveled, Potter?”
Harry Potter and the Order of the Phoenix, Chapter 17, Educational Decree Number Twenty-Four
A second’s relief, and then another burst of green light. The owl screeched and fell to the floor of the cage.
The motorbike zoomed forward; Harry glimpsed hooded Death Eaters scat- tering as Hagrid blasted through their circle.
But the owl lay motionless and pathetic as a toy on the floor of her cage. He could not take it in in, and his terror for the others was paramount. He glanced over his shoulder and saw a mass of people moving, flares of green light, two pairs of people on brooms soaring off into the distance, but he could not tell who they were—
“Hagrid, we’ve got to go back, we’ve got to go back!” he yelled over the thun- derous roar of the engine, pulling out his wand, ramming Hedwig’s cage into the floor, refusing to believe that she was dead. “Hagrid, TURN AROUND!”
Harry Potter and the Deathly Hallows, Chapter 4, The Seven Potters
In short Harry cared for Hedwig, it was just that owl's were low maintenance pets.