Is it recommended to keep my catalog on my computer's hard drive?

If you routinely use the "make a second copy" option in the import dialog of Lightroom Classic then that should solve the problem of forgetting to back up your images. Either leave your EHD plugged in, or set Google drive as the destination for your backup images. It works for me.

The procedure that I have adopted is that I automatically save a second copy of my images to Google drive when I import. My catalog and previews are on my main hard drive. I back them up weekly to an external hard drive. My working images are stored on other internal or external hard drives.

I have also experimented with having worrking images on Google drive. But in all instances the catalog and previews have remained on the main hard drive with backups on my external hard drive.

So, my question is: is there any risk in just keeping my catalog and previews only on my computer's hard drive and not backed up?

Yes, huge risk. This is completely not recommended. All important computer files need to be properly backed up.

In general, what is the suggested place for:

• Catalog & previews
• Catalog backups
• Images

You left off  "Images backup" from this list. The recommended place for catalog and previews is your fastest drive. The recommended place for catalog backups is any other drive (local or cloud). The recommended place for images is your slowest disk. The recommended place for image backups is any disk that does not store the originals.

This is apparently a continuation of your previous discussion from last evening. There is always a risk of having your catalog and previews on your computer hard drive and not having a backup. That was not my suggestion last night, and never would be my suggestion.

For best performance, my suggestion would be to have the working catalog and previews on the computer hard drive. The backup of those files could be stored on Google drive or on an external hard drive, but NOT on the same hard drive where the working files are located. Otherwise, if the drive containing the working files crashes, and that's where the backup is located, you have nothing to fall back on.

My suggestion in the previous thread was to consider putting your images on Google drive, leaving the catalog and previews on the main hard drive, and using the external hard drive to back up your catalog and previews. But even then you need to consider a solution for backing up your images.