As far as I can tell, slight lateral play is uncorrelated with accuracy. I have some barrels, both Contender and Encore, in which there is slight lateral play. Most of them shoot great for me; a few don't. I don't think it's the lateral play, but can't say for sure what it is. Good luck with yours.
On the other matter, what do you mean by tool marks? Nearly all new, standard rifle barrels will exhibit some tooling marks. Shoot it and see what it does. Also, my experience with T/C and their warranty has been positive. I'm confident they will replace your broken stock promptly.
As for your trigger, before you get too carried away, try dry firing some, or shooting some to get rid of those tooling marks. It will probably lighten up a bit. Try pushing on the rear of the hammer while dry firing. The extra pressure will burnish the mating surfaces and will probably lighten the trigger pull a bit. Don't overdo the pressure; you could chip something.
Like it or not, the reality today is that most new guns need a bit of tuning. Mass production keeps the prices down. If you want all the major manufacturers to have everything butter smooth, guns are going to be priced out of the reach of most mortals, because skilled hand work is expensive.
RonF