Yes, you can.

File a FIR, take attestation on photocopy from gazzeted officer and a affidavit that mentions that you lost your original.

Meanwhile, ask the school for procedures to issue a duplicate admit card.

yes you can attend the exam.

and you need not worry but i would suggest you to get the photocopy attested from a gazetted officer and give an affidavit saying that you have lost your original and that the photocopy may be treated as original. there shall be no problem if you get this done.

goodluck!