Will I be double-charged after reinstalling?
You will not be charged twice during the same billing cycle. Indeed, the Shopify system prevents that automatically since 2018, as you can see in the official announcement at https://www.shopify.com/partners/blog/developer-digest-december-2018#billing
When you uninstall the app and install it again, your subscription is canceled, so Shopify asks you to accept again the recurring charge associated with the app subscription. This happens even if you uninstalled the app one minute ago because your recurring charge was canceled.
Accepting a recurring charge does not mean that you have to pay at that moment; you will be charged in the future, in the next billing cycle, and only if your trial period is over.
NOTE: All payments are managed directly by Shopify. It is Shopify, not us, who charges you, and then transfers our part to us. Basically, we don't create the charges or control the billing cycle. We don't have access to your payment data either.