When you click on the button, then when you click on "walletconnect" a QrCode must pop up but nothing happens (it's the only feature that I want to do)

This error occur on the hosted github page to show the project here : web3modal.github.io/web3modal-vanilla-js-example so the example isn't working correctly, I can use it to my project

If you don't want to use Socket.io with your node provider for event you will need to request every block/event and your request's number will be too large so you will need a paid account for that. Socket.io can listen to it pretty easily. And if you don't want this, you need to be your own node.

You can broadcast them with RPC nodes : ethereum.stackexchange.com/questions/74713/…

I don't think you wan wrape "msg.senger" from an EOA account into a smart contract function. If you call a function with a smart contract, msg.sender will be your contract. Maybe if there is a way to "whitelist" your smart contract address so it can act like your EOA ?

Be aware that some libraries didn't implant replay attack so it will be ignored when you sign your transaction. And this transaction can be reject by some block explorer.