I can answer a few of your questions, those related to the dev process of the app, for the other, I guess you can do some research on the internet.
Would I use my stripe account that is already included in IPS framework?
Invoices and payments are managed by the IPS Commerce app, thats why you are required to own the IPS Commerce app in order to use my app, therefore any payment gateway will work ( even those custom ones from the market place ), although you have the option to select which ones are available in your merch store.
Does printful do their own billing? How Printful get their cut?
For printful you are the client, basically this is what is called drop shipping. You buy stuff from them, and sell it further at a higher price. So yeah on their website you can see how much the product costs on an invoice generated by them, that being the amount you subtract from the price at which you sold the merch on your website to find your earnings ( data already available along side each invoice in the app I created btw, also there are daily and monthly earnings statistics ).
So a selling process of merch on your website using this app would be something like this:
Someone buys some merch.
The app notifies printful of the order.
Printful generates the invoice for you and sends data about the invoice back to the app.
The app links internal invoice to printful's returned invoice, then tells printful to try and pay the invoice using the payment method available for your store in printful ( eg. You have a credit card linked to your printful account ). If payment fails or you have no payment method attached you will need then to manually go to printful's website and pay thw invoice before the process goes any further. If the invoice has been paid printful will do it's thing: print > pack > ship.
What happens if the customer has a Printful item and a specific to my site item in the cart at checkout? (A tee-shirt and a banner ad on my site for example)
Carts for this is is separate from the IPS Commerce app, couldn't make it work because of printful's shipping, but I will maybe try it again in the future as I might've been something.
So basically they have different "checkout processes".
Sorry if I coud not get do a better job formatting this post, or messed some up words, it's been written from my phone.