Translation:The artist goes to the cinema every week.
Without some context, using "is going" with "every week" is not possible
The sentence wouldn't be used in daily Hebrew either. The combination בית קולנוע can be synonymous with קולנוע when you're saying "going to the movies". However, if someone told me הולך לבית הקולנוע it would seem like an emphasis on the building (בית־), so I might understand it as going to the cinema building but not to see a movie—or think the speaker insists on using "high Hebrew" even though in this case there is no difference.
Just a term to describe sophisticated/flowery Hebrew, or a modern Hebrew infused with many old/ancient Hebrew words. It's not an official term or anything.