Hi diaritoch, the original Name field is provided by BuddyPress by default and cannot be removed as such. It should, however, only be visible on the BuddyPress User Profiles admin page, not on the front end. Is yours showing on the front end?
If you edit your BuddyPress profile, do the first_name and last_name fields in WordPress not reflect the changes? And vice versa - if you edit first_name and/or last_name fields in the WordPress user page, does your BuddyPress profile not change? Which version of WordPress and BuddyPress are you running? Just trying to get you to be more specific so I can perhaps reproduce the problem, which at present I cannot.