Does anyone have advice on how the process should work when you have multiple prospective tenants interested in a property? We posted a unit for rent and have received around 15 replies so far. Okay, a lot of questions here...

Do we ask everyone for an application fee and do background checks on everyone? Or do we just go in order from the replies we received? If multiple applicants meet requirements, is there a "legal" way in which you must choose who would get to be the tenant...like, do you have to go in order from the first person who contacted you who also qualifies or can you pick based on which applicant you feel would be the "best fit" based on qualifications such as income, professionalism when doing a showing, references, etc? 

Also, knowing there are so many applicants, what if someone offered us a higher rent to be chosen for the vacancy? Do we then have to inform all of the others? How would something like this work?

Thank you so much for any help you can provide! 

A little background: For our first tenant we had to find, we used a rental company (and paid them one month's rent) and never thought about all of this. For the second tenant we had to find, we heard from a couple right away who qualified and just went with them. We've never been in this situation before, but I'm sure some of you have and might have some advice?