Not (+34%) but (*1.34)
So if your final chance was 5% it becomes 5*1.34=7% (assuming it rounds upwards)...not 5%+34%=39%
It does make a difference though I saw that in my last battles fairly clearly.
I just delete the PDCs and put the new one in via SM. In general the changes are often dramatic enough it makes sense the crew training drops. Also they will over time regain their training just due to officers being shuttled through them.
Crew training also affects things like maintenance failures and repair times of battle damage I believe, I'm not 100% sure. In combat situations the training level of the formation is also fairly critical as every change you make to orders takes time to process. So things like changing direction can take time, opening fire can take time, etc.