Hi,
Thanks for the details, they were enough to find a bug on our side for one of the two emails.
When the payment method takes the money during the checkout, before the order is created, which is what the Stripe Connect plugin does when its "Payment mode" setting is on "Checkout", the payment notification email was not sent. HikaShop builds that email from the order, and at that point the order does not exist yet, so nothing was sent. We've made a new release of HikaShop today with the necessary modifications for this to work: the notification now waits for the order to be created and goes out right after.
The order status notification is a different matter, and it is expected in that flow. Since the payment is already accepted when the order is created, the order is created directly in the confirmed status: there is no status change, so no status notification. What the customer gets on an order which is created is the "Order creation notification" email. Could you check in System > Emails whether that one is published on your site? If it is disabled, your customers indeed receive nothing, which would match what you describe, and publishing it should give them their confirmation on card orders.
Two things which would help us confirm all this: the value of the "Payment mode" setting on your Stripe Connect payment method, and whether the coupon you tested with covered the whole order. If it did, no payment plugin ran at all during that order, which is why that test behaved differently.