If I purchase a domain name for the internet, does that give me the
right to use that name at, let?s say, a warehouse or store? In other
words, is/can my domain name be my actual business name in the US?

Maybe. Or maybe not.

If you acquired the domain name 'neme.com' (for example) and you
wanted to set up an actual business name of 'Neme.com Inc' (for
example) ...

You would still have to check that 'Neme.com Inc' was OK to use as a business name.

Simply purchasing a domain name does not give you rights to the
name(s) listed in the domain.  In fact, even though you own a domain
you may not be able to keep it if it interferers with another parties
trade marks.  I would recommend that you setup your company name and
THEN use that as a basis for your decision on the domain name.

-Tim

You still need to contact/regfister with your County Clerk's office
and register the business name. But, if it is already in use, you are
out of luck. For instance, if you bought the domain
"www.bobs-repair-shop.com" (or similar) then you wanted to open a
physical repair shop in your town called "Bob's Repair Shop", your ok
to do so unless Bob Smith from around the block beat you to it and has
a repair shop already named "Bob's Repair Shop". You can't have two
seperate existing businesses with the same name owned by different
people.  Unless it's a franchise....and that's a completely different
story.....

Rob in NJ