I consider using it as sort of a "brand-name", so it needs to have an "adjective feel". Such as "Thousand Hearts Films" (instead of "a thousand hearts films"). Google Translate gives "mille coeurs", but searching through the web I found "à mille cœurs", can I drop the "à" like I did on the English version? Is "cœurs" better than "coeurs"?
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.
For instance: 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.