I, too, have a Moto X and am having the same issues since the last upgrade. While I don't respond to e-mails from my phone usually, I do use it to keep an eye on my In Box activity then log in at home to make sure everything's on my laptop. I use the standard e-mail app to link to my Comcast e-mail account. I delete unimportant messages from my phone but want them to remain on the server so I can get them onto my computer later.
In the meantime, I'll make sure to not delete e-mails from my phone until I can find a good app. Thanks.