Be sure to have the rent receipts and other relevant document proofs with you if you are claiming HRA/Section 80 deductions in the tax return directly. They are not needed at the time of filing, but after your tax return is submitted and processed, the Department may ask you to upload supporting documents on the Income Tax website.
Yes, you can continue to invest in Section 80C even if you've missed the deadline to submit investment proofs. You can then claim a refund at the time of filing of ITR.
You will, however, not be able to claim allowances such as LTA and medical reimbursements.