Yes “Mille Cœurs” is the literal translation for “Thousand Hearts”. And yes, you can drop the “à” given by Google Translate, which adds nothing to your “brand-name”. The “à” would be used to link the thousand hearts to their subject.
a thousand hearts flower => une fleur à mille cœurs
In your case, if it’s a “brand name”, then “Films Mille Cœurs” is perfect (as opposed to “Films à Mille Cœurs”), but you loose your “adjective feel”. “Mille Cœurs” is not an adjective of “Films”, it’s a proper noun. If you need this “adjective feel”, then “à” is mandatory, but it looks like you loose your “brand-name” feel.
On the last point: “cœurs” is indeed better than “coeurs”, because “coeurs” is not correct (it’s a typo). The only and right way to write it is “cœurs”.
May i propose:
Mille et un Cœurs
In french we often add “et un” for expressing a multiplicity factors that shouldn’t be literally interpreted as exactly 1000. (i.e. When you wish to say “A Lot !” )
Just like the One Thousand and One Nights Tales which in french becomes:
Comptes des “mille et une” Nuits !
Perhaps better for your intended usage.