A tenant ask if he could pay rent on the 10th of the month when the lease starts on the first. Is this common or be allowed? Seems like he is asking for 10 days of courtesy habitation.
In most jurisdictions you have quite a bit of latitude in what you and your tenant agree to. Some people like monthly expenses to hit at a time other than the 1st because of the timing of their paychecks and/or other monthly expenses.
I take it as a sign they might be living paycheck to paycheck which is not ideal, but up to you to decide if you're comfortable with their financial situation.
If you do agree to a non-standard payment schedule you'd probably either have them pay 10-days worth up front and then a full month on the 10th, or a larger first payment on the 10th. Not giving them 10-days for free.
Unless explicitly stated that they want ten days free, this usually means they want to make a first, smaller payment (they pay on the tenth of their first month for the first ten days), or they make a larger payment for the first month on the tenth of the next month that adds the ten days. After either of these options you then revert to a normal monthly cycle paid on the tenth.
You can obviously argue back and forth if you dislike any of this, and depending on the contract and locale can probably play hard ball on it if you really don't want it, but run the risk of an angry tenant. Angry tenant's are basically always bad, as a lack of care of the property can cause you a lot more loss than ten days rent.
Seems like he is asking for 10 days of courtesy habitation.
Not necessarily. You can see it from different angles, and essentially, they match each other in the sum.
As others suggested, you might want to charge them ⅓ of the monthly rent upfront. But then, you have to consider that in the last month they live there, they only have to pay ⅔ of the amount due.
You might as well accept a continuous delay: they pay on the 10th, but for the current month. It might be the easiest way of seeing it.
Both perspectives are equivalent in the end.
Surely you will hold a security deposit anyway; maybe you can agree to a higher amount than usual.