All iPods output music equally so there is no real answer.
Yes, Beats by Dr Dre work on all iPod types
Yes, Beats by Dr Dre will work with any device that has Bluetooth.
Yes, Beats by Dr Dre work on all apple products
Yes, all smartphones, iPods, iPhones and iPads work support Beats by Dr Dre.
The overall best Beats by Dr. Dres are the Beats by Dr. Dre Pros but if you cannot afford them you can always buy the Beats by Dr. Dre Solo HDs they are very well functioning and they block all outside noise and for the money the Beats by Dr. Dre Solo HD's are the best.
Currently, the most expensive pair of Beats are the Beats Pro, which go for around $400.
Beats by Dr. Dre Wireless are the wireless Beats by Dr. Dre, below is a link to them.
Yes, Dr. Dre is the co-founder and co-owner of Beats by Dr. Dre.
You cannot get Beats by Dr. Dre products for free.
Beats by Dr. Dre Solos are the smallest over ear headphones by Beats by Dr. Dre.
Beats by Dr. Dre is a line of headphones co-owned by Dr. Dre himself.
Beats by Dr. Dre headphones are made in China.