1) Geeta the quote you copied has this line,
"You do not need to tell us about any extra jobs you take, and can do voluntary work in any sector. " the same sector/same level thing is a bit confusing, could you please clarify if you know more about this?
If I'm reading it correctly, I think you can do another job that is related to your Tier 5 job (i.e. you could do some work for another part of the same company maybe), but you can't take a second unrelated job.
For example, you couldn't do your internship and also work a few hours a week in a bar or a shop... any other paid work you do must be directly related to the Tier 5 job. However, you can volunteer unpaid in any area.
2) My sponsor is a registered non-profit, but I don't understand why I need a visa for 3 months for an unpaid internship--does this affect the non-profit's tax-exempt status or something? If US citizens can visit the UK for up to six months.... why is a separate visa necessary?
You need a visa because it is illegal to volunteer on a visitor visa. All work, whether paid employment or unpaid internship/volunteering requires a visa that gives you permission to work (a Tier 5 visa in your case), otherwise you will be breaking the law.
3) If I do actually have to get this tier 5 visa, can I do it once I arrive in the UK? I'm currently in Germany, how long would the process take? My internship starts on Feb 27, is this possible, or realistically should I tell my employers it takes a month? This all seems terribly complicated for a harmless 3 month internship....
You have to apply from your country of normal residence, so if you are considered resident in France, you should be able to apply from France, but if you are not considered resident there, then you will have to return to the US to apply.
First your employer needs to secure a Tier 5 Certificate of Sponsorship for you. Then when you have received it in the mail and you have gathered together all the required documents, you can apply for the visa.
Current processing times for Tier 5 visas in Paris are: 63% processed in 5 working days, 80% in 10 days, 90% processed in 15 days and 100% processed in 60 days (processing time starts after you have applied online, attended a visa appointment in person in Paris and submitted all your documents in Paris).
Current processing times for Tier 5 visas in the US are: 85% in 2 working days, 90% in 3 days, 94% in 5 days, 99% in 10 days and 100% in 15 days (the processing time begins after you have applied online, attended a biometrics appointment at a local USCIS center and mailed in all your documents to the British consulate in NYC).
4) Thanks for the tip about the BUNAC program, I'll definitely look into it. I just graduated from high school in June 2011 and deferred to a university (starting Sept 2012 instead of Sept 2011) so maybe I can whip something up.
More info on the BUNAC Intern in Britain program is here:
http://www.bunac.org/usa/interninbritain/