LongMen grottoes, Guanlin, ShaoLin temple, KaiFeng

Your plan is acceptable.
To Shaolin Temple, you can go to local railway station and at the east square, you can find the buses for the temple. The buses are at around 06:00 and 07:00. If you miss that bus, you can go to local bus station to take buses for Dengfeng where you can find frequent buses for the temple.
Outside the temple, you can easily find the buses for Zhengzhou.
It is good to stay in Zhengzhou on Day2.
From Zhengzhou Longhai Bus Station, you can find frequent buses for Kaifeng.
It is hard to tell you whether it is extensive to climb the Mt Hua. I know some old people with good health climbed that mountain. I think it depends on your body condition.
September is not a good time to see the Mudan flower. April to May is the best time to see tha flower. If you go there in Sep., I am afraid that you may miss the flower.

From railway station to Shaolin Temple, Direct bus only 06:00 and 07:00 ?

Yes, it is. When I was there, there are only two buses.
Since you will stay one night in LY, I suggest you to ask the bus station the day before to confirm, so that you will not miss the bus.