How to allow/setup crypto transfers/payments on my website? I'd like to learn how to allow end users on my website to pay for goods using cryptocurrency.

Questions I have:

1 - How would I confirm the transaction has been received ? 
2 - How would I confirm the value of the transaction is correct ? 
3 - Are there any API's I could use to achieve what I'm looking for ? 
4 - If I had to refund a user how would I do this ?
To check payment:
you can use blockchain api to do so:
https://blockchain.info/q

which is

https://blockchain.info/q/getreceivedbyaddress/[BITCOIN ADDRESS]?confirmations=6

confirmations can be any but minimum 6 is recommended

to accept bitcoin, I suggest u host a bitcoin core and use something like  Easy Bitcoin PHP Library to do so