You might have to define "within reach". From what I've seen throughout the rest of the text of the class, it means "within striking distance", or the five-foot area around the Duelist that it can swing at. That doesn't necessarily stop somebody from moving their full movement (even if that "full movement" is technically half movement), and then stepping through the Dimension Door, even if they take an attack of opportunity in the process. It doesn't even stop them from taking a five-foot step before doing so (which doesn't even provoke an attack of opportunity). Thus, the evil Wizard your Duelist managed to (against all odds) shatter through the defenses and perilous abilities of isn't barred from escaping... He just has to step out of the way first.

If "within reach" means "within a single movement range of the target", then the target can't escape the "within reach" range unless they have more than double the Duelist's movement, because the Duelist's Challenge cuts movement in half. I think this is more what you're going for.

Frankly, I like it, but I guess my understanding isn't clear on when Dimensional Anchor is applied, since "within range" is relative.
Okay, I guess I could clarify "within reach" as "within your threatened area"

And to respond to the Wizard 5 foot stepping and walking away at half his movement speed, remember the duelist can immediate action take a 5 foot step, and can make a tumble check to make that 5 foot step actually half your base movement speed. From there, you only need to have a little bit of extra reach, or a slightly higher ground movement speed than the Wizard, to be able to continue threatening him after he moves away. Which then forces him to have to spend a second move action AND be casting a quickened teleport spell to get away, or deal with casting from inside your reach and dealing with the penalties of that. So either the wizard escapes, at the cost of a massive action disadvantage, or the wizard is stuck. Either way, you did your job, though you may have trouble catching said wizard again without outside help.

While I can see this, and I wouldn't necessarily suggest giving Improved Uncanny Dodge to the PrC outright, a Rog4/Bar2 gets Improved Uncanny Dodge just from getting Uncanny Dodge from two different sources, whereas a Rog4/Duel4... Doesn't. Instead, they get an Uncanny Dodge they can't use with their armor.

I think it just makes sense from the perspective of a character that already has Uncanny Dodge applying their already intuitive knowledge to the training that they are receiving as they continue to progress, as base class combinations already do. It makes sense from a progression standpoint and is consistent with what Uncanny Dodge already does, and while I guess this isn't technically Uncanny Dodge, it does give some benefit to a dedicated duelist choosing Rogue over Fighter, since Fighter gets Uncanny Dodge from this anyway.
I guess I can see the logic here. I'll think about it, but I'll probably end up updating the class unless I can think of a reason not to.