I have some experience running dropshipping stores and also creating solutions for dropshipping stores, sharing my ideas:
1. I will know the prices and turnaround times for some items in the online storefront, but others will require an inquiry to my business partner in Japan. How can I handle items like this in my storefront?
I would add a note on the product details page with wording that they should contact you to ask about price and turnaround time, you can either include an email to contact, or link them to your contact form, set up an automatic forward of those emails if you want to skip a step of your team emailing the supplier.
For the price, since a price for the product is required, you could do this:
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Set any price
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Tag this product so you can identify if in the theme code (example “Variable Pricing”)
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In the theme code, for products with that tag, do not show the price (instead show the message), and hide the add to cart button (optionally replace with a link to your contact page).
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Create a custom product for that customer, with the exact price.
That last step hopefully you can skip, without knowing how the variable pricing works in your system or how often it changes, I can only guess but for example if the price changes every day, then you might have to do this manual method, if it only changes every once in awhile, ask your supplier to contact you when it gets updated, then update on your site. If someone orders during that lag period, explain that the price changed, and adjust the order accordingly, or if the price regularly changes up and down and it’s not a large amount, absorb the price difference to make it easier for the customer.
2. If a customer makes an order, how can I automatically send that information to my partner in Japan to begin work on fulfillment, without a “touch” on my end?
If your store is able to use Shopify Flow, you should be able to configure an email to automatically send to them based on the order conditions. If you don’t have Shopify Flow, or want more options, you can use Order Automator, this is a common use case for dropshippers.
3. Am I able to display prices in multiple currencies with real-time (or very close) accurate currency conversions?
I’ve used a few different currency apps, this one being my favorite: Best Currency Converter. You can ask them and / or some other currency converter apps to see if their function matches your needs.
4. If a customer is buying from my US-based site, is there a way to have the payment route directly to my supplier? Or does the money have to pass through my hands first? Trying to remove as many taxable touches as possible to keep prices at a minimum.
I don’t believe there is a way to connect multiple bank accounts and route payments depending on order conditions. I’m not 100% sure but can’t think of a logical way you could set this up, because Shopify would then be forming an agreement with multiple entities.
The most common form of billing I’ve seen is the dropshipper sends an invoice at intervals, for example daily, weekly, monthly (monthly is the best but requires a good relationship with you and your supplier, weekly is pretty common).