One of the best and most popular script used is Vbulletin. Most of the reputable forums are powered by Vbulletin. The second recommendation I'd have is phpBB. It is absolutely free and open sourced versus Vbulletin which is licence based. I have used both and had no problems with either of them. If you can fork out some money I'd suggest going with Vbulletin.

Here is a list of the biggest internet forums http://en.wikipedia.org/wiki/List_of_Internet_forums. You can see most of them are powered by Vbulletin and the second most popular script is phpBB.

Regarding fresh look you can simply hire someone or buy a Template.

Moot has had lots of good reviews. I have not personally used it, but it would appear that it meets most of your requirements. Your mileage may vary.

https://moot.it/

I'm not sure one exists. The general best practice is for businesses and communities is to go where your fans already are: Facebook groups, G+ pages, etc. the examples you mention are very Web 1.0 an quite clunky without lots of work. I'd be interested to see what the conversion of your fb fans into a stand alone community will be. Vbulletin and others are fine but, like I said, clunky.