eM Client does not seem to be reverse synching with Gmail. It downloads everything from my Gmail account, but my Gmail account online client does not reflect emails I have erased in the eM Client app on my desktop. Should it?
Normally, synchronization is done in both directions.
Are you sure that you have launched the sync after having modified your agenda.
Do you check your emails from another terminal (smartphone, tablet,…) ? Here again, you have to launch a new sync manually if it has not been automated via the application you are using.
That is normal behavior if you are using POP3 because the Inbox is not synced with the server. You can configure the POP3 account to remove messages from the server once they are downloaded or when they are deleted in eM Client. You will find those settings in the POP3 tab for this account in Menu > Accounts.
If you are using IMAP, and you delete a message in the Gmail Inbox in eM Client, but it doesn’t delete from the server Inbox, then there could be an IMAP error which you will find in Menu > Operations > Log tab.