As far as I remember from my inferno TA, constructs have their own damage, completely unaffected by whatever damage type your gun does. There are only a handful of doctrines that apply to your constructs (extra health and extra damage are the ones you will want to be using), but none of them modify the damage type of your constructs.

TA is fun, especially as it is the only pet class in the game and thus offers a fairly different experience. The additional customisation offered by the constructs and their modules gives you a lot of room to experiment, it's definitely a positive.

But I'm not sure it's the best class for seasonal challenges, especially for the timed ones. I had big problems with the timed VC challenge simply because TA has no real burst damage - your weapons do little damage (most of it comes from your constructs) and you have no way to instantly blast groups of mobs while running towards your objectives. It is more of a slow, methodical class, where your TA is safely hidden behind your constructs, uses weapons mostly to direct which enemies your constructs should focus and advances slowly, step-by-step.

There are ways around it, but they seem much more cumbersome and complicated compared to the other classes. Also, the constructs' AI is not very good - here's a couple of examples from my experience.

Don't get me wrong, the challenges are doable with TA, I just think it's a lot more hassle than with the other classes.

Edit: forgot one more thing - all the game enemies and events can be triggered by your constructs, which could cause you headaches. I remember failing a wandering servo skull because of that - my constructs were chasing some random mob and triggered the skull, which I did not even see from where my TA was standing. So watch out for that as well.