Most custom ROM's have the notification count functionality, but I don't know how comfortable you are rooting.

I'm not sure if there's an app solution, as I've always rooted.

I will say the 6P is the best phone I've owned. I have absolutely no itch to use anything else. I don't think you'll be disappointed

 

PhoneFusion's Visual Voicemail Plus app will show you a count of voicemails (new, old and ready for delete counts are all displayed on the apps icon).

 

Stock Android will not show notification counts on the app icon. You would need to use Nova Launcher and use their plugin for that. Nova is a good launcher and has a ton of customization options, if you're looking for something like that. It doesn't require you to root either.

 

Android does have emojis. I'm not sure what differences there may be between the iOS one s and Android.

There are a variety of different third party keyboards available on Android. I don't know if there is one exactly like the iOS one. Third party keyboard on Android tend to have more features than the ones on iOS and there are more of them. Likely you will not find one exactly like iOS and Apple does not make an Android third party keyboard.

Nova Prime launcher has that feature.

 

Android as of 4.3 has colour emoji, same as iPhone. So any recent device will be no problem.

 

Is it at all possible you are referring to Facebook notifications or perhaps
Yahoo mail?
Moving from iPhone to an android based phone is already a step in the right direction!
My biggest problem in current best phone is there are so many great phones quote choose from that I get confused with choice!
Good Luck,