When I am typing in an entry form on a website, the form becomes deselected, the typing cursor disappears, and I have to click on the form to type in it again. Sometimes it disappears right after I click on it. The other thing is when I open a drop-down menu on a webpage, it disappears as soon as I put my mouse cursor on it. I am using Windows 7, and this happens in all browsers. The issue occurs in Safe Mode and during Clean Boot, and with my security software (Norton Security Premium) disabled, the issue still occurs.

Hmm.. sounds like a similar problem I had a few years ago.
I'd say you should check if anything has been downloaded recently that could be interfering with the cursor. It could be that some software'sneed to update frequently, meaning they run in the background and always check online for any updates, which makes the cursor acting weird/slow.

One thing I can suggest is to go to Task Manager, and turn tasks off until you realize the 'culprit' to what's making your cursor act weird. Something might be interfering with your typing. If you turn of something and the mouse works, then you've found it. You should then, if possible, try to delete it. Or if you can't, just keep it disabled. 

If that doesn't work, then your problem might be because of the Windows Search service crashing all the time and restarting in the background. Try checking your system log for anything that fails and tries to restart. 

Another thing that you can try, is to go to the Control Panel > Performance Information and Tools and click on Adjust Visual Effects. There you will find settings like "Let Windows Decide", "Adjust for Best Appearance", "Adjust for best Performance" or you can selectively turn off features that you don’t really need.

This could also be a malware that's causing interference. Chances are fairly low if you haven't downloaded anything in a while, but it's always good to run a scan just in case.