Okay, I don't know why I didn't see this last week, but there are a number of problems with the statement you posted.
First, you have to pick which "FROM" you're going to use--when you type "FROM DISK", that indicates to the database engine that it should go out to disk to get the backup file. If you use "FROM VMDB2012$DM_DMPROD_FULL_20170830_020006", that is the backup device (which already has the path saved in the device). Don't use both; pick one or the other--and, since you seem to be having problems with the backup device, I personally would point to the .bak file directly and leave out the backup device.
Next, take out the "RESTORE LOG" portion completely, and do it separately. The first step would be to figure out if the .bak file is usable--if we get to that point, you can restore the log later.
Finally, do you "have" to move the database files? Or, can they be installed in the default directories? Everything you can remove to simplify this problem will help get at the root problem.
To start, please try running the following:
RESTORE DATABASE DMPROD
FROM DISK = 'D:\VMDB2012$DM_DMPROD_FULL_20170830_020006.bak'
WITH CONTINUE_AFTER_ERROR, RECOVERY;
After this command executes, if there are errors, please post them. If there are no errors, then you have proven there is nothing wrong with the .bak file.
Assuming the above restore statement works, you can then drop the database (to be able to restore the log file(s) you have) and restore the .bak and any log file backups. To do that, you need to alter the above statement and execute "NORECOVERY" instead of "RECOVERY". This will leave your database in the "RECOVERING" state, and you can then restore the log file backup(s). If you have more than one, make sure to use "NORECOVERY" on each of them, until you get to the last one...which you can then specify "RECOVERY".
After restoring the database, you can then move the files around using the following sequence (which comes from https://docs.microsoft.com/en-us/sql/relational-databases/databases/move-user-databases:
ALTER DATABASE DMPROD SET SINGLE_USER
ALTER DATABASE DMPROD SET OFFLINE
MODIFY FILE ( NAME = logical_name, FILENAME = 'new_path\os_file_name' ); GO
EX: MODIFY FILE(NAME = 'CORDDB', FILENAME = 'E:\SQL Databases\DMPROD\corddb.MDF'); GO
MODIFY FILE(NAME = 'CORDDB_Log', FILENAME = 'E:\SQL Databases\DMPROD\corddb_log.LDF'); GO
ALTER DATABASE DMPROD SET ONLINE
ALTER DATABAE DMPROD SET MULTI_USER