Mail synch between iPhone iPad Mini using IMAP

Question

This is a complex questions, so please bear with me.
Setup
I use Mac Mini desktop, iPhone and iPad. All less than 18 months old.
Mail is setup for IMAP
My mail is picked up from my ISP's server.
I use SpamSieve for spam, which puts spam in the spam folder "On My Mac"
On my iPhone settings are:
     Advanced Settings for Mail,
          Draft Mailbox > Drafts
          Sent Mailbox > Sent Messages > on server
          Incoming Settings: IMAP Pathe Prefix INBOX
Issues/Questions
Mail incoming shows up on all the devices, but when I send mail from iPhone or iPad, it doesn’t show up in sent folder in desktop
When I try and delete mail from the trash, I often get the message: "can't delete mail xxxx file is missing."
My list of accounts on the left pane have folders from my server, such as ".mailboxlist, .openwebmail, which I would like to see disappear.
The On My Mac section on the left has 12 folders like: "Sent Messages (sender accountname)"  I don't know where these came from, but nothing is more current than July.
I really do not grok the logic of how these settings should be handled for multiple devices and I haven't seen any comprehensive explanations on the net.
Mail seems to not pick up mail for extended periods sometimes and I have to close and restart for it to do so. The refresh mail button doesn't work until I do so.
I am too often rebuilding mailboxes, which seems to help.
Any help, explanations, suggestions would be appreciated by me and probably thousands of others.
Thanks.

Answer 1

Apparently you must be very narrow minded. I don't want to develop an e-mail client from the scratch.
I would implement the sync the following way.
Assuming I use POP3 on both my iPhone and my MacBook. On my iPhone mail client I am downloading e-mails and reading them from time to time and replying. There is an ended set of what you actions you can do with your e-mails. Every e-mail has a unique ID / fingerprint - you can recognize it with 100% accuracy. I'm also downloading e-mails on my MacBook mac mail and performe the same ended set of actions on this e-mails. So the system architecture should be as follows:
1. MacBook mail client queues all the actions I have performed with my e-mails on my MacBook.
2. iPhone mail client does the same - creates a queues with actions I have performed on my iPhone.
3. If I have the syncing turned on on both of my devices and I'm logged in on both of them with the same Apple ID both devices process the queues to my iCloud, compare it and update status of my e-mails according to the actions in each queue.
So I will rather wait until Apple will do this simple solution.