I know IF beat Cat in H4H, but personally I think they really nerfed his character in the final two issues by making him behave completely out of character and look like an evil Namor. Of course, my opinion doesn't really change the fact it happened.
However, more recently in Cable & Deadpool, Shen Kuei (Cat) managed to beat Deadpool pretty badly during their first encounter. He did comment about not surviving another battle with him, or vice versa, but in a fight between the two later in the same issue, Deadpool was unable to beat him and had to resort to taking a hostage.
In the same series Deadpool has squared off against Iron Fist twice, and without his iron fist, Deadpool matched him fairly evenly. In fact, he didn't seem to take the two fights they had seriously at all because of his healing factor. In their first confrontation he was busy proving Weasel's innocence, and in the second (where IF was under the guise of Daredevil), Deadpool literally stated that he was simply a "nuisance".
But the whole point here is that on a good day, Shen Kuei is on par with Shang Chi and on a bad day, he's just slightly his lesser.
As for IF dodging bullets, Shang managed to pull off some impressive bullet dodging maneuvers himself. In his MAX series, he managed to dodge several shots fired from behind him and no more than a meter away. Judging by the way he dodges after the bullets leave the chamber, it was definitely an on-the-spot reflex. There are also countless times where he deflects bullets using his bracers.
Now strictly between Danny and Shang, they have fought in the past where both were manipulated by a villain. Although both revealed they were holding back, Iron Fist lunged in with his "flamin' fist" at one point and Shang easily dodged it. There are also countless times Shang has been referred to as the best empty handed fighter/martial artist throughout his appearances in Marvel Knights, and even Black Panther unintentionally offends Luke Cage by telling him that "Shang-chi is the master."
Finally I want to bring up Iron Fist's brief scuffle with Hiroim of Hulk's Warbound. I believe Hiroim commented on Danny's strength when he used the iron fist, but nevetheless easily incapacitated him. Shang faced Hiroim as well in H4H and although didn't fair much better, actually blocked Hiroim's attack and at one point, the latter was using both hands to push down on Shang who was still standing his ground. Keep in mind Hiroim is a guy who other heroes have stated to be as strong as the Hulk before they uberly pissed him off.
Anyway, I'm just rambling on. Apologies for the massive post, but in all honesty I think that with the iron fist, both are pretty even. The problem is Shang knows about Danny's ability and would make it a priority to not let Danny execute the move properly. A lot of people just don't seem to respect Shang as much as other heroes and it's understandable given he lacks any distinct super powers, but it's still unfair.
Personally, I believe that Shang Chi might be superior, completely skill-wise, the technique that Iron Fist is mastering in his own series puts him over the edge in an actual fight.
and just as a side-note, does anyone know if the Iron Fist series ends after the Seven Cities of Heaven Tournament? Is it going to be an all out ongoing? I hope so.