There are closing costs associated with your loan. There are closing costs associated with the purchase including city and county transfer tax. There closing cost associated with title insurance. In a buyer's market you can ask for any of these to be paid by the seller or adjust the price accordingly. In my opinion it doesn't hurt to ask but be aware, the deal is best when it is a win - win for all involved.
Next time use an Agent so you Know exactly where you stand.
Yes, this is correct, unless you negotiated something different in your contract.
But remember that everything is negotiable in real estate, and if you haven't signed the contract yet, you can incorporate this into your negotiations.
Enjoy your new home.