Maybe you have some filter, content control, or spamhalter acting on the messages.

Suggestions to help you start to diagnose the situation.

In Pegasus:
Tools > Options
User Interface > Reporting/logging
Increase Reporting level to 5 - Debugging messages

Download Process Monitor, extract the ZIP archive contents and execute Procmon.exe.
Immediately stop the capture, clear the display, and add some filter rules.

File > Capture Events (Ctrl+E) or Magnify Glass icon
Edit > Clear Display (Ctrl+X) or Eraser icon
Filter > Filter (Ctrl+L) or Funnel icon
In the Process Monitor Filter dialog create filters:

Process Name contains winpm-32 include
Path contains .cnm include
Path contains .pmm include

Press the Add button for each rule.

Process Monitor Filters:

Press the Apply and OK buttons when completed.

Start a Capture (Ctrl+E)

In Pegasus; initiate a Selective mail download.
Select one message, Mark for retrieval, and Make it so.

Return to Process Monitor and stop the capture.

By looking at the capture you may get some clues.
Also look in Pegasus at Window > System messages.

Suddenly, the last few days, Pegasus New Mail Folder would not accept emails from gmail.com and would not allow stored Gmails from folders copied/moved back into the New Mail Folder. I see new Gmails in my email notifier and in Pegasus Selective Mail Download waiting to be downloaded. When clicking on "Check your POP3 host for new mail" all these emails would get deleted without downloading. Weird! Can someone please help me figure out what happened suddenly for this problem to occur? 

Is gmail the only email source or is it the only one exhibiting the problem?

If it is the only source, how many messages are in your new mail folder?  At some point above 700 the new mail folder stops being able to automatically display newly downloaded messages.  If this the problem, a refresh (close then reopen) of the new mail folder will display them.  If not, report back after restarting Pegasus and confirming the problem still exists.

It seems that the New Mail folder doesn't accept ONLY emails FORWARDED from my Gmail account, INCLUDING those that were saved in a folder. I don't have this problem with emails received from Gmail users sent to Pegasus or with any other emails, like Yahoo or AOL. And as I said, I do see these forwarded Gmails in my email notifier and in Pegasus Selective Mail Download waiting to be downloaded.

The New Mail folder consists of 55 emails and this is not the problem I restarted Pegasus and rebooted several times and this problem still exists.

Can someone please help me with the above problem why the New Mail folder doesn't accept emails from forwarded mail from my Gmail account, INCLUDING those that were already saved in a folder?

How are you receiving your Gmail messages?  You say "forwarded from my Gmail account" but forwarded to where?  How do they get into Pegasus Mail?  The norm is for Pegasus Mail to access your Gmail your via IMAP or download from your Gmail mailbox via POP3.  I am wondering where the "forwarding" comes in.

Might autofiltering be diverting these messages to a different folder?  To see a list of folders that have autofiltering enabled right click on any folder in your folder list then select Autofilter > Manage Autofilter folders.  Check this even if you don't think you have enabled any autofiltering.

You hit the nail on the head, Brian. Indeed it was the auto-filtering problem. Although the problem was not when right-clicking on the folder's Auto-filtering/Manage Auto-filter Folders. That was empty. It was in Tools/Mail Filtering Rules/Edit New Mail Filtering Rules. There I found 2 items, including my forwarded Gmail address, in "Rules applied when folder is opened", as well as several items, including my forwarded Gmail address, in "Rules applied when folder is closed". How in the world did they get in there when I never even touched the auto-filtering option? Thank you very much Brian for your quick response and suggestion. Good luck and all the best.

Glad you have it resolved.

Be aware that what you found is not autofiltering.  Those are filtering rules that AFAIK must be manually created, they can't be unintentionally enabled like autofiltering can.

But how did they get in there if I never created any and never even noticed this filtering option?

Good question.  The only ways I know of to create these rules are by using the mail filtering rules functions or by the manual text edit of a rules file.