Sql-server – How to bring a database offline safely

However, two days later, the .mdf file is the same size as pre-transfer (I took it offline and only the timestamp changed).

There was enough free space in your data file for any new data that was written. When that spare/free space is consumed the file will grow, not before. You can confirm the size and free space in your data file with:

SELECT 
    [File Name] = name
    , [Physical Name] = physical_name
    , [Total Size in MB] = size/128.0
    , [Available Space In MB] = size/128.0
        - CAST(FILEPROPERTY(name, 'SpaceUsed') AS INT)/128.0 
FROM 
    sys.database_files;

I checked the log, which is usually under 1GB, and found that it has now reached 30GB.

Your database is in full recovery mode and you aren't backing up the log. For this and all manner of transaction log growth explanations, see @MikeWalsh's canonical answer on the topic at Why Does the Transaction Log Keep Growing or Run Out of Space?

Attempting to repair the database isn't going to help. Page 1:9 is a system object and system objects can't be repaired. You'll want to script out the tables and create a new database. Then create the tables in the new database, and copy over the data into the new database.

If the database won't open you can try setting it to emergency mode using ALTER DATABASE MyDB SET EMERGENCY which may bring the database online. If you does you can script the data out. If not you're basically screwed.

If you had a backup... Another option would be to restore the database to a point in time before the page became corrupt. If you have a backup from that far back you can restore that backup, then roll the transactions logs forward and you should be able to get back to a state where you have all the current data without the corruption.