While the treatment of Anderson at the end was heartwarming, there's an element of practicality to his decision: four corruptible judges were killed in the course of the film. Given the high crime rate, and manpower spread thin, they need every possible good judge available, especially one who is not only a powerful psychic but more idealistic.
Screw heartwarming, she's more than earned it. Sure, she got captured by one of the perps, but she then proceeded not only to wipe the floor with the bad guys, she also saved Dredd's ass. She's clearly more than competent. Toss in her psychic powers, her idealism and the crying need for judges, and they literally can't afford it to fail her.
Additionally, I read the end as Dredd being impressed by her pressing on and finishing the bust despite firmly believing that she's already failed. Being a judge is about more than being fast with weapons; the corrupt judges are perfect counter-examples.
I also think Dredd was doubly impressed that Anderson dispensed justice by letting Clan Techie go and getting him to back down over it. Again, being a judge is about more than being fast with weapons.
Really, passing Anderson was both principled and pragmatic.
Dredd constantly asks Anderson questions about what to do next. This is not just explaining to the audience (because he's Dredd, of course he knows what to do next) - he's testing Anderson and asking her examination questions while they go about their business. He even evaluates her - there's a lovely scene after they break into a Slo-mo den and Dredd pop quizzes her on sentences and procedure after they've pacified the situation, and Dredd gives a tiny, almost imperceptible nod after she tells him the right answers.
Also notice that Dredd usually makes her walk in front — which makes sense because this is her evaluation, so he's keeping her in his field of vision so he can watch how she does. However, Dredd moves her back and takes point whenever they're breaching a door — which also makes sense, as when you're going into a room full of bad guys you want the first guy through the door to be the best marksman.
When Kay fantasises about shooting Anderson during a Battle in the Centre of the Mind, he's imagining himself using a Lawgiver. Guess he really did always want one of those...
Why wouldn't Kay know that the Lawgiver would backfire on him? Because judges don't go to Peachtrees and the gang has had no practice against them or their equipment.
Also, the technical details of Lawgivers are most likely classified. Living in Mega City One, a citizen will probably see a judge use their weapon at some point. Kay even states that he always wondered about it, adding a Shrouded in Myth reputation to its capabilities. If the self destruct mechanism is common knowledge, then enterprising criminals would try and reverse engineer Lawgivers to disable this (well, after the first few are blown up anyway).
Something similar actually happened in the comics: a gangster managed to get a hold of the Lawgiver chips and changed them for others that would make the new Lawgivers explode if they were used by their assigned users.
Further, it's quite plausible that too few perps who capture a Lawgiver live to tell the tale. Just ask Kay.
The vagrant just might have saved Dredd and Anderson from blast door squishing by distracting them just long enough before the door closed.
Anderson is psychic, so why didn't she know at the end that Dredd was going to pass her?
Her powers are not passive. She needs to concentrate or at least try. She didn't.
Helmets interfere with psychic powers!
She might not have wanted to find out that way. Some sci-fi psychics refuse to read others' minds without permission.
If Ma-Ma's bomb is rigged to go off if it her heart stops beating, wouldn't the transmitter going out of range cut off the heartbeat signal, thus detonating the bomb?
No, the switch sends off the detonation signal if her heartbeat stops. Going out of range is irrelevant to that.
The helmets explicitly interfere with her psychic abilities, how was she able to read the female Judge so quickly and easily?
The helmets interfere with her psychic abilities if she's the one wearing a helmet.
In the comics, one ability Dredd does have is that if he concentrates, he can resist psychic probing. Since this film is quite true to the source material, it's pretty safe to say that he can do this here too.