From the pictures it looks as though Hazel is a Dane, or at least mostly Dane. I'm sure the vet already cautioned you about too much strenous work at her age - large breed dogs and heavy joint exercise in youth is a baaaaad combination. Just make sure her body is mature before you start having her do any serious work.
As far as the distractions, I find "watch me" a very good cue to teach young, excitable dogs. My collies are intense enough I don't need to, but my aussie would be after a rabbit like a shot back in the day, too. A very effective training method I used for my aussie, actually, was based around schutzhund obedience and I think it works incredibly well for dogs that are easily distracted, although it might mean going back to the basics with your Hazel. The essential part of the training is you never teach the dog commands like "stay" - that is he is instead trained to do what you have told (sit, lay down, whatever) until he is given a release. Regardless of what you do. Regardless of if you move, walk away, or if it's 10 minutes or a full hour or more, he follows the last command he was given, until he is either given a new command, or released from the last. There is a lot of conditioning, and it takes longer to train this way, but it's ultimately more effective in the long run, and you might find you have good results with your Dane on this - and probably your Weim too.
The most serious work we do are small jumps. Now that summer (and the heat) has hit jumping is restricted to evenings. She does love to run, climb, and rough house with Duke, I don't usually restrict her in how much she frolics about. She has, of course, access to clean water 24/7.
I've been working towards the 'stay/sit/lay down until released' with Hazel. Her top time currently is about five minuets when I've been actively training her. She's gone longer when just given a command around the house, but I've only timed her during training sessions (normal session lasts around 10-15 minuets, often times twice a day).
The dogs I seen pulling were MUCH heavier built. Again, the first dog is a Dane mix. Personally I really would not recomend using her only becuase of the breed's very common hip issues.
I would consult your vet about her for sure if nothing else.
I've actually seen German Shepherds and Rotti's used to pull. Hazel's taller than a Shep and can exert as much, if not more, force as a Rotti. She's checked out completely a-ok with the vet. I don't plan on doing any real pulling until she's more in the 2 year old range (I don't think she'll be ready for any real pulling until then as it is)
I'm looking for plans to really well balanced carts. There was one gentleman built carts where per 100 pounds placed on the axles exerted 1 pound of pressure on the dog.
Generally draft dogs are Swissy's, Bernese, or Bernards, but there is no reason any breed can't do it as long as you are not overloading them in the weight department. I would worry about Hazel's young age, however.
Like I said, real pulling wouldn't be until the 2 year range or so. The most she'll ever be pulling is in the 120-130 pound range, and always on concrete or pavement, never on dirt/gravel/grass. Her feet are pretty tough, but I'll probably have to look into buying or making her some pooch boots to protect her paws. Not just because of hot surfaces, but also because of rocks and glass.
I don't intend on working her in cart every day, or for long distances either, more when I can't/am unable to walk or run her because of my back or knee or on a cool day and I have nothing better to do than take her on a ride around the neighborhood.
You should see her when I take her on hikes through the woods. Occasionally Duke will get a ways ahead of her and she'll take off like a rocket. It's amazing to watch her, it's more like she's flying then running. Every 2-3 three seconds you here this lil 'pumf' of her paws hitting the ground. From sniffing in a bush to Greyhound in .2 seconds! XD
Most of the time she passes him in the narrowest breath of an inch, but sometimes she runs smack into him 'cause she misjudged a bit, or he moved into her path. Hurts like mad when she plows into you. She's clipped me only once in the several times she's gone super-dogging past me.