72 hour free transit

Actually, you will be eligible for 24 or 144 hour visa free transit and not 72 hour visa free transit that has been replaced in Beijing by 144 hour visa free transit a couple of years ago.

Having separate tickets is fine and also traveling on unrelated airlines is also fine. Now, you can calculate by yourself whether you will be eligible for 24 or 144 hour visa free transit by counting your time from your scheduled arrival in Beijing, until your scheduled departure from Beijing. In case if your transit time in Beijing is less than 24 hours, you will be eligible for 24 hour visa free transit and if your entire transit time is longer for just a minute than 24 hours, you will be eligible for 144 hour visa free transit. In reality, you are fine in both cases.

You should print out your flight tickets and have those printouts with you and you are also advised to book your single overnight in Beijing in advance and to have a confirmation printout with you.

Separate airlines are allowed. Though not recently, I've heard stories about Emirates incorrectly denying boarding if not on a same ticket.