Arya is riding side-saddle in front of the Hound facing his left side, but when the scene angle changes (as she brings up wanting her own horse), she is now facing his right side, even though they haven't stopped riding.
When Daenerys speaks to Missandei on the cliff, her gown flaps in the wind. In the very next shot however - taken from the opposite side - it hangs completely still.
The process shown at the beginning of the episode is called casting, and cannot be used to make swords - they'd be too brittle; forging is a much more elaborate and lengthy process. If the secret of the steel is lost, as both Tywin and Jaime mention, it is impossible to just melt an old sword to make a new one. It is necessary to cut the old sword into parts and reforge it, and to know the exact method (i.e. "secret") to do that, and even then, it'd be a totally different steel.