Reservations cannot be made on the app, you can use eurail.com most of the time though. There are extra fees so it's always better to book through the railway company instead. Most European trains also have optional/no reservations.

I do not recommend waiting until you get to the stations : some trains could be sold out.

If you want precise advice mention your itinerary with travel dates if possible. Check timetables on bahn.com or DB Navigator app (German Railways) : they're connected to the European-wide database and are more reliably updated than Rail Planner.

You only need to log the train right before boarding so by this time Rail Planner should be correct. If not, add the train manually.

Paper passes are not included in the sale : however you could exchange the promo pass to a full fare pass for a 15€ fee + price difference. Take into account the 1-2 months delivery time to the USA. I wouldn't do that honestly.

You'll get the gist of the app quite early I'm sure. I've used Rail Planner during a month trip and never had an issue.

I'd recommend reading those helpful links :

https://www.seat61.com/how-to-use-a-eurail-pass.htm

https://www.seat61.com/interrail-and-eurail-reservations.htm

Feel free to ask further questions.

Thank you so much for your information and advice. 
It is very helpful.