Dear Sir,
1) It is their fault. You are still eligible for HRA exemption. Rectify the rent agreement for future claim.
2) It doesn't matter.
Keep lease/rent agreement with you showing same amount of rent for which you are claiming HRA. Also make payment by account payee cheque and if rent amount exceeds RS.50 lacs, deduct TDS @5% on it in the last month of tenancy.
Thanks