If you don't have a Form 637 nobody is going to charge you 10% on the sale of your product.
If you don't have a Form 637, you will have to pay 10% when you buy your components if you buy from a supplier that collects Federal Excise tax.
Yes and No , the Form 637 allows you to buy components tax free IF the store has a Form 637, most likely a retail store does not because the tax is paid at the manufacturer level, that is where is supposed to be paid. If you adding to the component like painting or assembling, then you are a manufacturer. then If you sell at retail, you have to pay 10% of your selling price.
If you have established wholesale pricing, and regularly sell your product at this wholesale price, then when you sell at retail you only have to pay 10% of you wholesale price , not retail.