Would you buy from a shop if you had to register in order to make a purchase? Im debating removing the need for registration. This has pros and cons in my opinion. Pros - People are more likely to make a purchase as your removing a barrier and making it easier. Cons - You dont get peoples email addresses etc so you are unable to market any promotions to them at a later date...potentially losing business in the long run.
Don't keep registration, just give them an option to put in email while checkout, usually people would input the right one.
You can send newsletter to them later as long as the client agree to received the newsletter aka opt in list.
The more steps I have to go through, the more likely I am to give up and buy from somewhere else. I'd suggest just asking for the email on the checkout page so you can send them an invoice electronically, then have a checkbox they can tick if they want to get other mail from you as well.
Registering what you are buying? For me it is depends on what you are going to purchase, if its need registration, why not.