They show if they are capable, same way they showed 3G & 3G+, HSPA & HSPA+ (now seen as H & H+ respectively)
This is also dependent on what the network chooses to push out. In Kenya you will primarily see 4G but abroad it's both 4G/4G+ & LTE. Lumia users should see LTE on supported devices.
While more power efficient, initial implementation of 4G in Kenya was a battery nightmare no matter where you were. Had a Lumia 1520 & iPhone 6/6S/6S+ all LTE capable but they kept overheating and restarting. Battery life was 2 hours standard on all phones (this was in Late 2015)
Distance from the signal tower/signal strength where you are is a key factor. In the CBD, my phone on 4G (with location turned off) manages about 5 hours of continuous online use - FB, IG, Whatsapp. In Ruai, It managed 3 hours under the same conditions because I had 2 bars that kept dropping back to H+ (This was On Jan 1st 2017)
This is mostly true. You can get the advertised 5-7 hours battery life on your phone while using wifi. Then again, factors like signal type (a/b/g/n), frequency (2.4Ghz vs 5Ghz) and distance from the router/obstacles in between will matter.
I've managed 5 hours use time on FTTH in the last month, sitting 10 metres away from the standard router they give you.