I went to the company which prepared taxes for me and filed 1040. I discovered that as a F-1 visa holder I should have submitted 1040NR. Who will be responsible for this mistake?

In the eyes of the IRS, YOU are ultimately responsible for filing the correct tax return.

However, you relied on the tax professional's expertise, so he should file the amendment to correct the error at no charge to you PLUS, if there are penalties and intetest, he should pay them.

If that is not happening and you need professional help, look up my email address on my profile.e

Thanks for your response and help offer. I'll consider it.

Yeah, but what about the amount they already charged (it was approx. 3% of estimated return)? Do you know (or heard) any potential consequences for the Green Card application in the future if the mistake is not corrected?

I should have filled 1040NR-EZ and instead I filed joint return of form 1040 and we also got a child credit. I'll be in a trouble to collect that amount of money. Can I use credit card to pay back to IRS (hopefully it will be only child credit).

Yes, you CAN pay whatever is owed to the IRS using a credit card, but there is normally a 3-5% fee from the credit card company.

You can also set up a payment plan with the IRS.

Thanks, answers are extremely helpful.
Should I worry about the amending the state taxes as well? If not what implications it may have on my current status and on possible green card application in the future?

There is probably NO NEED to amend the state returns, because the states do NOT differentiate between U.S. citizens and foreign nationals the way the U.S. government does.