Nothing will be charged from Shopify end for publishing/reviewing apps. 

Here is quick idea, not a full code.Create a folder 'front-app' and place all your react app code here.And add <div id="react-render"></div> in liquid teamplate, where we need to render the componenetsfront-app/index.js   file will be like import App...

It depends on the themeIn dawn theme, it was under the file buy-buttons.liquid  

Fields mut be inside the <form> tag. Add the fields before closing the form.Also wrap name with properties[]  

Can you please share your site url, So I can have a look

You can use Product Options App for this requirement.Try our app https://apps.shopify.com/product-options-addons Checkout this example created using this App:  https://product-addons-demo.myshopify.com/products/community-example-2484658

You need to wrap the field name with properties[].Change the field names as followsname="properties[Your name]"

In mobile it will show on clicking the plus icon towrads the menu.  But in your site the code I shared in the screenshot has repeated twice.So, the scripts is not working as desired. You can check your theme file where the custom codes added and try ...

The code block in the screenshot has repeated, I can see it in two times. It is supposed to be run onceTry removing it.    

It depends on the tasks you planning to do. What are the tasks you need to achive? 

Are you using this template https://github.com/Shopify/shopify-app-template-php/.Or any other, please specify, then can suggest the required changes to be make to connect with MySql instead sqlite