To the OP,
For any 3rd party email app, I have found that you should leave it in the background for it to work best. Don't swipe up and completely close it when not in use or it won't download the emails in the background.
Spark has worked great for me in the past but I am not a fan of threaded emails and with Spark, you don't have a choice. It's threaded all the time.
I was using the iOS mail app and had my home emails forwarded to my Outlook.com account for push access on IOS, but the idiots at MS broke my connected home account when they updated the Outlook.com website and can't get it working again.
So I started forwarding my emails to my Gmail account but since the iOS mail app DOESN'T do push for Gmail, I had to find an alternative.
That is why I started using Airmail. It's $5 but has quite a bit of customization. Still a bit buggy but much better than it was.