I can't understand your problem. I have just tested a Samsung Galaxy S4 GT-i9505 Android 5.0.1. Settings > My Device > Language and Input > Samsung Keyboard > English (US) > Qwerty Keyboard > Check what other settings you require > Return to Desktop. Tap Messaging > Tap Write Icon > you should see your keyboard and above the keyboard (on the left) is the Emoji Icon. When I tap on that, the lower screen fills up with emojis - faces - objects > animals > buildings > symbols - in fact 760 available emojis!!!
My phone is the Samsung available in New Zealand and is unlocked, unrooted and uses the standard Samsung Version 5.0.1 Android. If your phone is carrier supplied (Vodafone, TMobile, Sprint etc.) it may have dedicated software without emojis but I find that hard to believe.
Have I mis-read your problem? You said you found that your phone is Version 5.0.1 Android. When you open the phone to Desktop, does it show a screen with Time and date at the top and below that a white bar with Google on the left and a Microphone icon on the right and Apps icons at the bottom of that?
If so, Tap on the Google word and your Qwerty keyboard should open. At the bottom of the keyboard in the middle is your Space key. On the right of the Space key there is a key with a Full stop (Period) and a small ? mark with 2 dots (..). Press and hold the key and a menu window will open with the word Sym on the top right. Tap that and an emoji icon should show on the key to the Left of the Space bar.
Tap that emoji key now and you should see the full availability of Emojis a total of 154 in the Google mode. I hope this explains what I have on my Samsung and that you have the same.