Will low power mode influence my facetime connectivity (speed and stability) or whatsapp messaging system? Namely, I have an iPhone 5 with a very weak battery, and I will form now on probably hold it in low power mode at all times. I am wondering if whatsapp messages will take a beating from this, and also if facetime will lose its speed and stability during the calls. Any suggestions, advice from experience guys? Thanks!
Considering according to Apple the low power setting affects 
"Background app refresh
Automatic downloads
Wi-Fi associations"

By reducing or disabling these features,

Then yes it will be affected, your WhatsApp won't update its content unless you have it open as the app you're using. FaceTime should still remain as good though on cellular networks but if you're moving around while connected to a wifi network you may have performance issues if you stray too far away while low power mode is on.

Source: https://support.apple.com/en-au/HT205234

Also further info on back group app refresh here: "Use Background App Refresh" https://support.apple.com/en-au/HT202070