Menu item path saved as node/nid instead of alias
Both nid and aliases are not guaranteed to be stable and unchanged, nid being a bit less prone to mutation. What you seem to really need, are UUID Links. It should allow you to store links by the unchangeable UUID, not localised nid.

There was also a sandbox project UUID Menu for just the case, but it seems it was never properly initiated.

The other approach is to export both content and menu as a Feature. This problem is in Features' issue queue and in UUIDs for Menu Links issue queue, unresolved yet. But in the second one patches exists.