Is there any way I can control the Dragon, and the Direction he perches? Sometimes he is perching backwards before I one cycle him, so I can go onto the other side. But sometimes he just turns after 1 Bed and I fail the one cycle. Can I do anything about this?

No, but it's not that hard to look up see which way it's turning and go to the correct side of the fountain to one cycle

If the dragon turns after you bed it, it usually means you just did it a couple ticks too late. I also struggle with this. All you need to do is practice one cycling more so you can bed at the perfect time.