do I need any agreement for renting room? I mean from tenant not from landlord/owner.

private house no , just make receipt for deposit.
Renting from offices (housing , apartments ) Yes .

hi facebook, private house no?
will this work as proof of residency when applying for citizenship?

proof of residency generally refers to letter mailed to that address by bank or government or state . so i don't think you need agreement papers.

i mean proof of continuous residence.
to show continuous residence we must provide agreement or rent receipt.

Your I-94 stamp will show when did you leave US and for how long. You will fill the dates from there in the travel history section. You don't need to prove where did you leave in US during the 5 years residency. Just fill the dates where did you live and from which date till which date.

written agreement is always better than verbal agreement....especially when deal with money