I just signed a lease for a rental house. I am instructed to send all rent payments by mail.

According to the lease, the standard rent rate is $995. However my landlord agreed to give me a discounted rate of $950, as stated in the lease, as long as my rent payments were RECEIVED (not postmarked) by the 1st of the month. The lease agreement read that rent is considered late if received after the 5th, in which case a late fee would be imposed, of course. This means there is a five (5)-day grace period before rent is considered "late," right?

The other confusing part is that the lease states that if the rent is ever RECEIVED (not postmarked) one or more days after the 1st, even if it's received before the 5th, my discounted rate of $950 is still forfeited nevertheless, and the rental rate defaults to the standard rate of $995 for the remainder of the lease term.

My question is this: Is it lawful for a landlord to have such a stipulation in their contract that allows him/her to discontinue a discount for rent and increase the rate even if a payment is received before the so called "grace period" ends, as defined by the contract?

Yes, this is completely legal and not uncommon.

I do not understand why you are having a problem, they give you an extra discount if you pay before the 1st, this make sure they get their money early and ontime.

This is totally different than the 5 day late fee where an extra amount is due.

So if you are paying after the 5th, you lose your discount, and the rent goes to the original amount and you will owe a late fee.

Completely normal and happens all the time, in fact my current lease has a similar clause.