What about Google contacts? It does sync changes from device to Google? I don't want to move my data to icloud.
Google contacts and calendar will sync to the iPhone without a problem: just add the google accounts in settings and enable the options for contacts and calebdars. Your paid account will support push in the native mail account if you add it as an exchange account.
For the others you need to add them as a google account and that doesn't support push email: if push email is important to you, you can install the gmail app (and just disable the email option for the google accounts in settings so you don't get them delivered to gmail and the native mail app.)
Also, if you have more than one calendar on a single account, check on this page
https://calendar.google.com/calendar/syncselect?pli=1 to make sure they're setup to sync
You can also install the google calendar app if you prefer it to the native one (if you do this, disable the calendar option on the google accounts in settings to stop then appearing in both)
I ran my google account on my iPhone for 18 months without any issues at all (I now use office 365 though)