I think you should stay away from draw reins; getting a headset right now is the least of your worries.
For the stirrup irons, run them up like this: http://www.geocities.com/gerrypony/images/stirrups.jpg
There should be NO excuse for irons flopping around like that. It's unsafe - if you can't be safe about it, then just take the reins off the bridle.
When she changes directions in a round pen, she should stay on the outside and swing her front end in, do a turn on the hindquarters, and switch direction that way, not run through the middle of the pen.
A SURCINGLE is this piece of equipment: http://www.croftequestrian.co.uk/images/4171.jpg
HAF, choose ONE thing to perfect before you move on to the next thing.
It seems to me like you "choose" one thing to work on, then the next week you're changing your mind and wanting to rush ahead.
If you're going to work on groundwork and manners, then STICK WITH IT FOR A MONTH. Don't think "okay well I worked her in the round pen for 2 hours this week, great she's fixed!" and move on, because it doesn't look like she's "fixed." She needs to be doing things consistently - I.e. Not tripping over her own feet, not turning her butt to you to change direction, not coming into your space.
That means that you definitely don't need to even think about worrying about draw reins and headset; that's a long time off for you right now.
If you're going to do Parelli, great - STICK WITH IT, FOLLOW IT THROUGH - don't change your mind every other week! As far as I know, draw reins aren't a part of Parelli...
I still stick by my opinion that she looked lame on her left front; if your vet doesn't think so, get a second opinion - she was not sound.