I find the throw hand pointer finger to be more problematic on the long run, drying out on the thumb side. it made me actually wear a throw hand glove.
the non throw hand pain will go away once you build callous skin (like on fingertips for guitar or bass players), but if you live in a wet environment (sea side or a country like Japan for example), using gloves will definitely help (also make sure you pick up appropriate strings as some are more vulnerable to wet atmosphere than others)
about the trapeze and double or nothing and landing those.
I find most beginners tend to try and move their fingers around too much. Your fingers are not actually supposed to move in order to do the trick, you need to have them in the correct position, be parrallel to the yoyoing plane and just position yourself, hands, fingers or body around the yoyo and the string. You don’t actually put the yoyo on a string, you just move yourself in a way that the yoyo ends up going where you want it to go, relating to you, not to itself.
Most of these moves are very small adjustments which is why it seems that the yoyo player actually makes the yoyo go here and there while he just is moving the string configuration around the moving yoyo. At least for such tricks as trapeze or double or nothing. But it’s the basic idea.
You should be able to land a trapeze easily without using your finger, try using an horizontally held pole, or umbrella handle or whatever that you can easily keep horizontal. You can also try with a friend’s finger, just tell your friend not to move at all and you can throw and do a trapeze with his finger just being there. Double or nothing works the same, only with an extra loop over your throw hand. (make sure the first loop around your non throw hand finger is rested against the back of your finger, it will give you more room to end the double or nothing using the tip of your finger and it will keep the first loop far enough for the yoyo not to land on both strings)
As for the throw, I found that working slowly, taking time to decompose your movements, making sure each move is perfect, will do a great deal with improving your throw quality and also spinning speed/time. Technique is more important than strength when it comes to throwing a yoyo, an experienced player will get more spin time just letting the yoyo slide off his hand on a strength free throw than a beginner throwing with all his strength.
for me, I must admit that I never was comfortable using the traditional 1A breakaway, and I saw great and fast improvements in my side throw/breakaway once I started to practice 5A; The 5A hold (you hold the yoyo with your thumb and pointer finger rather than the inside of the palm) feels more natural to me, there is a wrist flicking motion there that made the whole thing click for me. Maybe this would be something to explore, later down the road.