Yes, and No.
The _vti* files you see are part of the extensions, but they may have
been copied from your previous site. If this is the case, the
extensions need to re-installed, and followed by a server health
check. _vti* files are unique for each web site and each server.
There are other bits of the extensions stored outside of your web.
Looking at the file structure with FTP will not *normally* damage the
extensions, but manipulating any files or folders with FTP will
corrupt the extensions.
The FTP server, and HTTP server are two different things. There is no
need for them to have the same UserId or Passwords., so if one works
for you it does not have any bearing on the other.