Hi Joe,
happy to give you some feedback as an European...
1) I suggest using Interrail for transportation, Europe is well connected by train and Interrail should give you the freedom to explore the continent at your own pace:
https://www.interrail.eu/en/interrai...t-is-interrail . Of course we do have many cheap airlines, so look into RyanAir / EasyJet / other low cost carriers for longer distance flights (e.g. if heading to Portugal / Spain).
With regards to your travel pace: I have travelled for an extended period in the last year and I really loved to have the freedom to stay in a place without tight schedule. Also check out smaller cities as DZ suggested - Interrail should work well together with this.
I would recommend going by bike only if you are travelling in warm season and if you have more time for a smaller area.
2) Stay in hostels and you won't have any problems connecting with other travellers. Especially as an American.
Most hostels offer single rooms as well if you need privacy, but still want to use common facilities.
3) I find your list quite Northern Europe-heavy. Giving your travel times (October) I would suggest only doing Iceland and one place in the Scandi region before heading further south to get some better climate. I would absolutely throw Portugal / Spain / Italy in there - for me these are absolute top destinations in Europe due to having great culture (very outgoing, life takes place outside, history), excellent food and nice weather.
If you want some great alternative / urban culture I would throw Berlin in the mix, for great historical buildings and a generally nice city Munich.
If adding some Southern/Western countries to your choice it should give you a solid experience but 5 weeks are definitely not enough to see it all. Also, let other travellers give you some hints on what to check out, especially with regards to smaller places.
Enjoy your trip!