Organizing Mail

Hello,
I am curious to hear your advice regarding how to best manage my email, of which I receive a ridiculous daily amount, making it hard to stay organized.

My company has our email hosted through Gmail, which I have recently started accessing in my browser on Zoho Mail. I started using Zoho Mail because:

I do not care for the flat label system in Gmail for managing emails and much prefer using folders
I do not care for an Inbox that is seemingly NEVER empty in Google
I use Zoho CRM on my Mac and Zoho Mail is the best (only Mac way?) way to push email into CRM Contacts

Before discovering Zoho, I used Apple Mail which I loved because it offered the ability to move emails to folders stored locally on my Mac. I also have an iPhone that I use heavily when away from the office, which also pulls flat labels from Google in to appear as folders on the device.

The trouble is (or rather, ONE of the issues), neither Apple Mail nor Zoho Mail work with Gmail to empty the Inbox.

After spending hours today sorting through my Gmail on Zoho Mail, moving emails to a nicely organized file structure and deleting useless emails, the deleted emails are still showing up in my Gmail Inbox as if I had done nothing to organize them.

So I suppose my question is: once I've pulled emails into my new Zoho Mail file structure, are they "there"? Meaning, if I organize everything in Zoho and then go to my Gmail Inbox and permanently delete everything, will the emails remain in my Zoho Mail file structure? Also, does CRM pull from my Zoho Mail version of my Gmail account (with folder structure) or does it pull from Gmail directly? On my iPhone, is it possible to access my Gmail account through Zoho with the organized folder structure?

Furthermore, is there a nice and tidy way of tying all of this together? Ultimately, I'd LOVE to use my Apple Mail as my sole go-to to access email (along with my iPhone), as I prefer the flexibility of a local folder structure, as well, although this is not necessary. (I do believe that there is no support currently for sending contact info from Apple Mail to Zoho CRM.)

Thank you for reading and thank you for your advice.
Melissa