Abandoned carts is a constant problem with any e-commerce site. In fact, about 50-80% people adding something to the cart abandon the purchase according to industry standards.
If you can give me a link to your store and your specific problems with abandoned carts, I'll be able to help out in a better way.
In general though, it is helpful to think about this problem from the point of view of the customer and their stage in the purchase journey.
Why are they abandoning the purchase? If they have added something to the cart, we can assume they did find something they were looking for. So the reasons for abandoning now could be:-
To recover these abandoned carts, it becomes important to engage with these people who are abandoning the website. This retargeting can happen through multiple channels - Facebook Ads, Google Ads, Email, SMS, Messenger. Ideally you should optimize for channels which allow you to get into a conversation with the customer if you have not yet nailed why are customers abandoning your website.
I am actually building out something to solve this problem (check my website) and I'll be happy to help you :).
Is it me or is the ability to send an abandoned cart reminder through the Shopify platform disabled after the 1st reminder is sent?
My Abandoned Cart settings in: Settings->>Checkout are set to the recommended "10 hours"
No, I don't think there is a way to send more than one reminder from Shopify again after the first time. You'll have to use an external app to achieve that. If you want to do that specifically from within Shopify, you can look at this app called Seguno which lets you do email marketing from within Shopify admin dashboard.
Yes, you can go to orders and then abandoned checkouts. You'll be able to see all the correspondence history you are looking for.
Let me know if that helps! :)