1. Some have found the rubber gators around the front shocks get torn, so I'd look at those. Check for even wear on the tires...could need some new tires and an alignment. Check the vehicle status menu and see when the various services are due. If any are overdue, there should be a red exclamation point on the lower left of the display as well.
2. Checking the actual battery capacity is not as easy as one would hope. Note, the range to empty is not a fixed value based on battery capacity...it's a reflection of how the car was driven in the last 18-miles in comparison with what it thinks is available, and that will change based one where, when, how, and the temperature of your driving past.
3. There's a lot of controversy on this. On an i3, when the display says 100%, it is not really fully charged. The logic just won't let the batteries be fully charged OR discharged (but when it gets to an indicated zero, it won't move...doesn't mean it's really out of power, though).
4. Yes once the car is registered to you, you can then gain access. It will let you do things such as honk the horn, flash the lights, lock, unlock, check where it is, turn on the cabin conditioning, and when you set a departure time while connected to charging, prewarm the batteries for maximum range when you leave. You can also access that info with a browser rather than that app.
5. The range to empty is a constantly varying calculation based on how you were driving...drive at speed on the highway, or up a long grade, and it will expect you'll continue to do that. Start out slow in a city, it will expect you to continue that...when you change from one condition to the other, it will gradually flush out the old history and keep updating with a blend of what you're currently doing. The community seems to average about 4-miles/kw. Some are quite a bit less, some more. If your commute was 75-80 mph on the highway, you'd be using more than someone going through town with a speed limit of 30-40. Live on top of a hill, you'd get great increase going down, but coming home, it will drop much faster.
2014 i3 BEV, 2021 X5 45e
(The i3 will be sold soon, <17K-miles, interested?)