If you are selling goods B to C goods overseas then these would be exports. If you send the goods to a non EU country then any import duties should be picked up by the importer overseas and so there would be no requirement to register for VAT in the UK and no futher action would be needed by you in the overseas country,
If you sell goods B to C to the EU and the goods are over £135 then you have a choice as either being the importer of record in that country or letting the consumer be the importer of record. If you choose to be the importer of record then you would pay the EU VAT and you would have to register for VAT in that EU country and account for the VAT on the sale to the consumer in the EU.
If you choose not to be the importer of record in the EU then the importer in the EU will have to pay any import VAT and there will be no requirement for you to register for VAT either in the UK or the EU.
If you sell goods to the EU below £135 then you can either register for IOSS in the EU and pay the VAT on one IOSS return for all the countries you import to within the EU or you can let the importer pay the import VAT in the EU.If this is the case then again there would be no requirement for you to register for VAT in either the UK or the EU.
If you sell to a business in the EU then the IOSS system doesn't come in to it so again you have the choice of either letting the business importer being the importer of record or your vusiness being the importer of record.
Again if you are the importer of record you would pay any import VAT and you would need to register for VAT in that EU country but if you ket the business importer be the importer of record then there would be no requirement for you to register either in the UK or in that EU country.
EU VAT e-commerce package
Please contact our imports and exports team regarding your questions about the value of the goods you disclose on the exports and the return of the goods to the UK where there are special import procedures.
Imports and exports: general enquiries
Regarding your question as to whether the gallery would be required to pay the import VAT in the EU would depend again on who is the importer of record in the EU country. There will be special import procedures in that country which would likely allow them to defer any import VAT until the time that the goods are actually sold but the gallery in the EU would need to look in to this.