You should be standing and moving in proper solid (re: anchored) stances rather than 'shuffling' about like a modern boxer. Your footwork should more closely resemble the way your legs move during katas, and I'm sure you don't shuffle in katas. :) A common mistake some people make is to stand 'on' their stances rather than 'in' them. The centre of mass/weight should be distributed in a way that it effectively "anchors" or "sinks" the practitioner into the ground. The common analogy is to imagine that you are a tree and that there are roots shooting out from beneath your feet and attached to the ground. This should allow you to better absorb incoming force, or quickly recover if you are upended (e.g. by quickly switching to another stance to regain your balance, or worse case scenario, performing a breakfall).
Now in saying that a person should be rooted in their stance, it doesn't necessarily mean you have to be rock-solid like a statue. One should be solid but fluid at the same time. This is something that I find a lot of schools don't teach properly, the idea of being simultaneously hard and soft. I find all too many people are either too 'hard' and don't move fluidly, or they're too 'soft' and they move with a good flow but insufficient solidity.
Some misinterpretations I've encountered include:
+ You switch between being hard and soft - e.g. when throwing a punch your arm is relaxed (soft), but hardens up on impact (hard).
+ One part of your body is hard, while another is soft - e.g. you are hard from waist down in a solid stance, but soft from waist up as you're punching/blocking etc.
I don't find either explanation to be terribly good, because they seem to view hard/soft like a light switch which means that the two properties are mutually exclusive. They seem to completely miss the point of the Yin Yang which shows hardness and softness being mutually inclusive, and the two dots at the largest portions of the opposites showing that hardness can be found at the height of softness and vice versa. I prefer analogies like looking at a whip or a crashing wave at the beach -- things that are soft in nature but also extremely 'hard' and strong in the amount of force they deliver.
http://www.calasanz.com/strong-martial-arts-stance/
So we want our stances to be strong and solid, but at the same time not "dead." They should also be agile and mobile... but not too mobile that you lose strength in the stance. It's all about striking that balance. :) There are various different exercises that you can do to practise this. I personally find Tai Chi's Push Hands is quite good.
http://www.youtube.com/watch?v=5r59gWTzKSw
^It looks deceptively gentle, but each forward push is actually producing a good amount of force. The object of the activity is to actually shove your opponent off balance with every push. A single mistake can send you flying backwards. :o
As for the groin strike, that's got more to do with your defensive posture. At virtually no point in a fight should your groin be exposed to an opponent. But we all make mistakes, and unfortunately stuffing up your groin defence is an extremely painful one to make. :o