Igst invoicing to foreign client, right or wrong?

Hi

My firm is struggling to get clarity on IGST invoices to foreign based clients?

My client is based in USA and payment currency is in dollars automatically converted & get credited in INR to my firm’s bank account. My firm is based in India.

Now

1) Is it correct to send my client IGST included invoices?

2) If they pay that IGST included amount and my company CA files IGST then later who can claim that amount my firm or my client?

3) Technically my Client is paying IGST so even if export of goods/services is zero and still he is paying then he should get refund, Isnt it? in that case what my client needs to do?

Thanks in anticipation,
Mayur