that's not a correct way. google services won't work smoothly.
before update the google play services, you need to solve the error notification first.
don't forget to go to play store, and disable auto-app update, so google play services won't update itself to newest version. your screenshot of no.2 may causesd by those update.
check if your google play services version is 22 now, if so you can restart, there will have uninstall button.
install google play services version 17, and follow these step:
-Turn off wifi and data
-Go settings to App-Google Play Store-click force stop store, go storage and clear cache then clear data
-Go settings to App-Google Services Framework- go storage and clear cache then clear data, repeat clear those for 3 times,
force stop until gray out
-Reboot device (no internet yet)
-Go settings to App-Google Play Services-clear cache, delete all data, then force stop until gray out
-Reboot, turn on wifi, go play store to check
*if error remain, try it a few times