Does Saitama still continue his training? Saitama mentioned that he got this strong through daily training, which consisted of 100 push-ups, 100 sit-ups and 100 squats along with running 10 km every single day. Also, he ate three meals a day, but only a banana as breakfast and also didn't use the AC, to strengthen his mental resistance. 

Has he stopped his training and accepted his current power level as sufficient? Or does he still train using his methodology? (I've read till Vol 6, Chapter 32 and haven't found anything about his current training status)
As Eumel's answer points out, in the manga nothing says whether he is training or not. However, I think it is safe to say that he is NOT training anymore, as the power he obtained is clearly too great for him to have any "exciting" fight.

The picture below, shows Saitama defeating Vaccine man in the first chapter. He is clearly pissed off that his fight ended up in only one punch, indicating that he would much rather have a more even match against a foe. Once realizing that he was clearly overpowered compared to the monsters he fought, there is no reason why he would still want to train. Even more, he might want to slack off hoping that it would make him weaker, yielding to more exciting fights for him.