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My payment page>Send payment (I find it a bit confusing as I associate generating an ivoice with receiving a payment although one need to somehow send the invoice to the payer in order to get paid.
no, you just need to choose LN, generate an invoice from your LN wallet and paste it there; also do remember to claim the leftover fee with cashu, as it will overcharge a bit more sats at first.
As for your questions - from what I've noticed, the default goes to the minibits mint, but you can also set the default mint yourself and then choose to auto-swap to it.