This is Dallas from the Shopify Social Care Team.
The payment gateways that are available to connect directly with your Shopify Admin are limited to what is available in your country. If a payment gateway is connected to your Shopify admin then the sale and the funds will be automatically added to your Shopify analytics, you would see all of the information about the order and all of the payment details in your order page, and you would process any refunds from your Shopify order page. May I ask where you're located? I can get you a definitive list of all of the different payment gateway options that are available to you in your country.
If you're using POS, however, there is a bit more flexibility. When you're making sales in person you're actually able to use whatever POS system and payment gateway that you want to use. I just want to make sure that I clearly state the different features that will or will not be available depending on what route you choose.
Situation 1: If you have a third-party POS system with a third party payment gateway that you like then you're certainly able to use that, but it'll just mean that the order and payment information won't be automatically populated to your Shopify Admin. In such instances, you would have to manually enter your information into your Shopify Admin to show what orders you have made. That could be done after the fact by using the Draft Order feature.
Situation 2: You use the Shopify POS system with a third party payment gateway. If you were to use Shopify POS with a third party payment gateway then you would be able to create an order in the Shopify POS app, but when you go to charge your customer you would just click on External Terminal rather than Credit Card. You'll process the card on the third-party terminal and the funds will be routed through them and won't ever be in connection with Shopify. In the Shopify POS app, you just state what type of card your customer paid with, then just click Charge once the funds have been captured on the side of the gateway which will then process the actual order on Shopify's side.
Once that order has been processed an order form will be created in the Shopify Admin for you. Easy Peasy.
With that scenario, however, I just want to let you know that if there was ever a need for a refund you wouldn't process the actual refund in Shopify. Instead, you would go to your third party payment gateway where you would refund the actual money because that gateway has access to the fund information. On Shopify, you would just mark the order as refunded so you know for your records.
I know this information can be a bit confusing, but this should answer your questions. If you have any more questions at all please let me know.