I'm simply saying that push mail on my iPhone 5 is unreliable. Doesn't matter if I'm looking at iCloud (which I'm using basically as a forwarding service for my personal email account) or the connection to the Exchange server. Both crap out every other day for hours at a time. And yes, iCloud is worse with that.
So far nobody could explain to me what the problem is. Not the "geniuses", not the phone company nor anyone else for that matter.
Let me give you an example how annoying this is: I have a dying family member at the moment and tried to stay in touch long distance with family while in a meeting. Only to find out two hours later that text messages (regular not "iMessage" (the names alone drive me nuts) had not been delivered to my wife's phone. After that several calls dropped and were garbled. And that in the middle of a big East coast city with great coverage on T-mobile (maybe it was them but I doubt it).
I'm tardy to the party but I respectfully say that I don't see examples in your post that clearly point to the iPhone as the issue. So you don't think I'm blowing smoke, among the other things I do I'm an Exchange and SMTP e-mail administrator and I used to manage mobile devices where I work including the Vendor relationship with a carrier. Big whoop. What the equipment is (iPhone vs blah) is uninteresting to me (meaning I don't defend the Apple logo).
I assume you have more details but this isn't a good place to get technical support so why bother.
In regard to the texts, unless you subscribe to a premium service (if available on your carrier) they probably won't be traced or delivery confirmed (depends on the carrier YMMV). You mentioned talking to your phone company but I'd call them back and be persistent that they are "losing your texts" and see if they can bump up your service or take more interest. You need confirmation that the texts enter and exit their network (unless the endpoint is on their network). It's been my experience that the iPhone is good about notifying me in a timely fashion when it can't send a text so the assumption is the carrier received it.
SMTP mail can be delayed or intentionally throttled, especially with free e-mail providers. Instead of iCloud, I forward my personal Yahoo e-mail to Outlook.com because to it the iPhone is a true ActiveSync client. My experience is that it's more reliable than Yahoo. iCloud is, as they would say on Seinfeld, "Breathtaking". Keep a copy of the e-mail in all inboxes so you can compare the time it was received by your personal e-mail provider, received by iCloud, and try to make a note of when you were notified on your phone.
I'm rereading all this and not sure I'm being much help. How you connect, what you connect to, what it connects to, your phone, your carrier, your location... It all makes a difference.
If you haven't done so lately, remove all the apps from your recently used apps list and reset (reboot) your phone. Also make sure you have the latest iOS 6 or iOS 7 version on your phone. iOS had some pretty nasty Exchange bugs in early versions of major releases. Get a coworker to send a text at the same time to the same recipient as you do if yours are regularly not received, or also send a text to a different recipient. Bleh.
Crazy stuff. Good luck!