First the stepping stones don't auto tile correctly(first image) which I'm unsure how they're meant to.
That's why I asked you to show how it's set up in the editor or tilesheet, because the images that Avery posted are not formatted as plug 'n' play. It's a good idea for you to check out what the expected file formats are for anything that you want to start replacing - in the editor, click Help -> Contents -> Documentation -> Asset Standards.
You can see from the diagram of the A1 tilesheet there that the stumps in Avery's download are placed in a block on the sheet earmarked for waterfalls, which have their own format. Visually, you can tell the stumps are actually using the other regular autotile format.
If you cut and paste them up a column, into any of the blocks labeled C in the diagram, they'll autotile properly.
Second, when I use the water auto tile, it flickers the stepping stones (as seen in the next image) when the water is on it's fourth moving loop it brings all these stones in.
I don't see this in my test game, but I bet it's related to the point above. Animations have no fourth frame. The two water autotiles from Avery's download work for me where they're placed.
Otherwise as seen in the next image the grid seems slightly off placed which makes it double the work to build a map
I also don't understand this part - are you saying the white outline in your picture is what you see in the game? The A5 sheet is the correct size and alignment for me, I dropped it into MZ and it looks perfectly fine. Perhaps try re-downloading the file? Are you using any plugins that do anything to the drawing of the map?