Can I put the database inside the NAS or is it better on the computer desk?
The catalog database should be on the fastest drive possible. A catalog database stored on a NAS is almost certainly not going to perform well., but the program won't stop you from doing it. If you want to try it, you could always move the Catalog later.
What exactly are those yellow rectangles? Do they indicate how much time is left until the cataloging is complete?
They indicate that the task to the right of them is in progress. If something is wrong they will be red. If there is nothing going on with that task or the task has completed. They are not an indication of how much time is needed, just the status of the task. If there is an estimation of the time it will take to complete the task, it will say it (in this case it says 21.6 hours for the Catalog metadata gathering step).
PM now, it seems to be very slow. Is the reason why it is previewing images?
Any of the tasks listed are going to be somewhat I/O intensive, and any other tasks that need the drives that it is scanning will now have to share the drives bandwidth with the catalog tasks.
If I were to turn off my computer, will the catalog resume once I reopen PMP?
It will.