Hi,

The omission of surfaces and access tags on small segments is sloppy editing on my part, which I apologise for.
As for the separation, if it's just a line of cobblestone (although I'd imagine it's sett, as I've never seen cobblestone used in such cases), it's easily traversable and thus not physical separation. The path should therefore be combined into one linear way, with tags describing the different surfaces. 
One way is almost always nonsensical for foot traffic (except on footways in e.g. airports). [The community is pretty aligned on this.](https://community.openstreetmap.org/t/poll-interpreting-oneway-yes/123638). Furthermore, exceedingly few bike paths are actuelly one-way, legally speaking, but that's another issue. 

Would you be okay with me fixing my slip-ups and leaving it at that?

Best regards and happy mapping,
Hidoo