They are two separate things and I would recommend to have both if you can, as Like button is get people to share your home page with their friends and Like Box is to get them to join your Facebook page which is important have more followers for retention and branding purposes.  From SEO point, it is important if people share your page but you also need to get people to share your future posts which is also important and to do this you need to make sure you grow your followers on Like Page. Like box on home page would be good way to get more followers.

I hope that helps

I had that argument with a client once and i manage to persuade them not to go ahead with it as they had .com and .co.uk .

.com is international so i think it did make sense not to use .eu but in your case, if you don't have .com then probably you need to look at which countries you want to target. If your client is UK based and if they target client from all around EU, then it might make sense to use .eu as you have very little chance to target someone in Italy with co.uk or .com.fr domains. If you are targeting only UK and FR then, you don't need .eu. It will just duplicate your work.