New iPhone 5c Mail won't push

Question

I activated my new iPhone (Verizon 5c, 16 gig) on March 7th and updated to 7.1 a day later. It replaced an iPhone 4 running iOS 6. I had the phone all set up with programs and wireless connectivity prior to activating it. All of my emails and contacts were on the phone and they worked exactly as they had on my old phone using iOS 6, via my wireless modem.
However, either after the activation, or the update to 7.1 a day or two later (I don’t know which), my two personal email accounts connected to my two web sites stopped working as they had before.
My Yahoo and iCloud accounts did not change. They work fine.
There are two problems. First of all, I do not have the choice of setting my email to Push. The accounts can only be set to Fetch. I have contacted my provider, and they have assured me that there is no reason for that to happen. They have made no changes, and my iPhone emails almost always arrived on my old phone before they arrived on my computer. They were definitely being pushed, and I am also certain that they were being pushed before the phone was activated, because again, I almost always heard the new iPhone beep first.
To clarify, I do not have the choice to select "Push" when I go to the Settings>Mail,Contacts,Calendars>Fetch New Data. I do have that choice with my Yahoo iCloud accounts.
The second problem is almost stranger. The emails ONLY show up after I open and look specifically at those accounts, and they show up as already read (no little blue dots next to them).  So I don’t know that I haven’t read them. They still show up as unread on my computer. Also, even though they are not read, they also do not show up in the unread number account. (I sent numerous test emails as I fiddled with settings and this happens every time.)
And though the Notification Center is are set identically for all four email accounts, only the Yahoo and iCloud accounts provide notification. The two problem accounts do not notify me when new emails were received, even after the 15 minutes Fetch time has elapsed, and they still don’t show up in my email list until after I open that account and wait for the new emails to start showing up. They are never there before I open the account.
So, I can’t set the accounts to Push, I’m not notified when they arrive, and they show up as already read. (And by the way, those same emails still show as unread several hours later on the laptop because they have never been opened on either device.)
I know for sure that I had Push notification before, because it irritated me that the phone always seemed to get the message before the computer, and in my business every second counts (and I can respond faster on my laptop).
Note that when I am home, I have cell phone service due to my location. But I do have wireless connectivity for emails and internet. So it is not a phone carrier problem. I still have the same problem when I am away from home and on the Verizon network. The emails still don’t show up as they should.
If this can’t be fixed, I can get my old iPhone up and running again, except it has iOS7 on it now (a friend bought it, but will give it back if necessary). Is it possible to reinstall iOS6 on the old iPhone 4? I have to be able to get my emails faster.

Answer 1

Update
I was wrong about the notification. Going to POP made the emails that arrive show up as unread, but it still doesn't notify when the emails are fetched.
So fetch is set up for 15 minutes (I need it to be faster) and when it does fetch (and I'm not sure that it does) it doesn't notify me. Only when I open up mail do I get the notification sound telling me that there are new emails.
When I'm traveling, I need the phone to check mail for me. And I need it to tell me that it has new mail. Why is this so hard?