I'm not a lawyer, so I can't specifically address legality. And I'm not licensed in D.C. However . . .
It is permissible to accept a higher offer for rental property. I was looking at the MLS (MRIS in the DC area) just yesterday for a house I'm going to be renting out. It's in Falls Church--quite desirable, overall. And I saw a couple of comps where the actual rental amount was above the listing amount. It's also a pretty hot market--average days on market is around 8. So, yes, it's permissible to offer more than the asking price.
You ask whether you have the "right" to expect that applications are processed and considered in the order THAT THEY WERE REQUESTED. Probably not. In fact, you probably don't have the right to expect that applications are considered in the order they were received. A landlord can pick and choose from among applicants, just as a seller can. And so long as the picking/choosing isn't based on some legally discriminatory element (race, religion, national origin, etc.), that's entire legal and entirely permissible.
I'm surprised that you received information on other applications submitted. But that's in your favor. Now you can apply and adjust your offer to make it more desirable (and more likely to be selected) by the landlord. If you have a Realtor, ask your Realtor for advice.
Hope that helps.