Should we registered our company name OR our own name on social media site? Why?
I would use your businesses name if it relates to your business activities because that’s what visitors and customers of your site will look for, but if you are a freelancer and you’re just using it to get a bit of extra publicity or awareness (and to make friends) then you should use your own name as it’s a personal endeavour. Essentially, if it’s for business, use the business name, if it’s for personal use, use your name. 
Yes, you should use the your businesses name on social media site for share business activities and help to improve brand reputation.