I'd say they're too tightly wound at the head and possibly sharp...I change mine myself and have never had that problem.
Take it to a guitar shop and have them put the strings on and see if they break. If they do, go to a guitar maker/repair and have him take a look.
You could try a different brand...but it's unlikely that that's the cause.
If by "into the tuner" you mean the tuning pegs, I'd say the edges are too sharp on the pegs. Soften them down somehow. I'm not sure how, to be honest; I'd consider taking it in to a guitar shop somewhere.
Yeah, that's what I meant. And I'll talk to my dad about taking it into a guitar shop somewhere.
You could try a piece of highish grit sand paper, high grit shouldn't cosmetically damage or do damage to the guitar. Just enough to make it slightly round. I wouldn't do it though, it's always better to have a pro do it not to mention cheaper than a new guitar.