To answer your first concern:
Your database file (the MDF) likely still has free space from it's initial default size. A database usually has free space inside the file it utilizes. Eventually, it will fill up and then you will see it get larger (assuming you have that enabled). Until it reaches that point the size will not change, even though data is being saved to it.
As far as the log file goes:
The fact that you're asking this question tells me you are not very familiar with SQL recovery concepts. Do you need up to the minute recovery? If you have a catastrophic failure, will you be restoring the last full backup or will you be reapplying transaction logs?
The simplest solution to your problem if you do not have the ability to properly manage or apply transaction logs is to set your database to SIMPLE recovery mode. You can then basically forget about the log file (it will stop growing out of control).