Thunderbird Portable.

Your 2nd Trash folder is a Local Folder.
You can create and delete Local folders based on what you want to keep Locally.
You can move/copy emails from a Gmail folder to your Local folder to keep a copy of the email.
Move or copy emails from the Gmail Trash folder to the Local Trash folder and see if the items count changes.

I use Tbird Portable with IMAP accounts.
I use the Unified View, i like that best as i only care about incoming messages from any email account.

In the Menu bar click on View > Folders and try the different options listed.
See what you prefer.

Here is snip of my Unified View:

ICIT2LOL said:

Forgive the ignorance but just what is portable?

This is basically what portable is in a nutshell,

A portable app is not installed and as a rule, does not put associated files and configuration information on the host device but stores all application files and data together.

Because files and data are self-contained, portable apps run independently of the host operating system

What is portable app? - Definition from WhatIs.com

Roderunner said:

Hi DavidE, this is what I get using Unified Folders. I use POP3 for my Gmail account although I did have AOL using POP3 but I deleted the account.

With only 1 email account, Unified probably wouldn't be much help for you.
I like Unified because i can see all Inbox's for 3 accounts in one list.

Callender said:

Another tip: Right click each email folder and choose "Compact" as it helps to keep things running smoothly. Do this on a regular basis - say once per week.

I never manually Compacted folders.
I let the Options / Disk Space settings take care of that, hopefully it works correctly.