I carry a couple of cubes, have a couple of old ones at home that I'm slowly salvaging for parts when I need them. The standard in both my main cubes is 50m Skyline or Zing It, I used to really like the Marlow braided line, but after multiple dodgy ones (broken within a week) I stopped using it. There's usually a 10 or 12oz Harrison Rocket on the throwing end of the lines, a 16oz weaver on the tail end. I also carry about half a dozen other bags ranging from 6oz - 16oz.
I have a Bigshot, but rarely use it. I can throw pretty high quite accurately. I favour a one handed throw, without any of that basketing gubbins, just a load of faff imo. Ivy covered/dense trees I'll use a heavier weight to blast it's way through.
To join the bags to the lines, I'm settled on bowlines girthed onto the ring. I don't really see a need for slips or dropper knots. If I think there's a chance the bag will get stuck then I'll let it run to the floor and take it off before pulling the line back through. I probably throw an average of 5 shots per day, this year so far I've not got a single bag stuck. Last year, probably less than 10 stuck bags. An old krab is great for giving that extra tug on the line when you need to unstick it, just pop a Munter hitch on the top and the bottom and use the spine as a handle.
I throw nearly every time I want to set a rope, either anchor points in trees, pull lines, traverse lines etc.
Throwline games are one of my summer past times Take the dog for a long walk and see what I can get the bag to do, walk it up and down the tree, multi tree redirects etc. I'm also partial to a bit of boomerang throwing, I have a nice scaffold bar set 3ft off the floor for playing with.
Imo, the throwline is an underrated part of our kit. I'm fortunate enough to work with some groundsmen who can set pulls in limbs and set rigging ready for you to get to the cut, massive time and labour saver when it's required.