A car does not normally leak oil (this is the year 2016, not 1951), and in any case oil does not belong on any surface (certainly not where there's foot traffic).

In the end, none of the stains came out.So the question is, in your situation, what would you do as far as penalizing the tenants such as by deducting something from the deposit for the stains (seems to be located in heavy traffic area where kids would be sitting where the couch used to be) on the new carpet and the stains on the old carpet?

It took the plumber 15 minutes...but saved me two hours in traffic.

Your manager should be re-marketing the property right this minute and yes if you are afraid you weren't getting a lot of traffic before then you should be providing some type of incentive (lower price or move in special) to get more interest and get qualified tenants fighting over your property.

For the rest, I'd make an appointment for them to see the unit, letting them know that the first approved tenant who actually signed a contract and paid the security deposit got the apartment  & that I kept showing the unit until that is what I had.And if they were late (without a call with a dang good reason like they're stuck in bumper to bumper traffic, which I could verify on Google maps) or didn't show, they were denied.  

This might change over time - as you get into one class of building/tenant you might discover you love it more than you could have imagined or you might figure out you'd rather navigate mall traffic at Christmas. 

Secondly, do you have a special lease agreement that allows for you to take a larger than normal security deposit due to the heavy traffic and higher chance of property damage?

I used to do that, and noticed that when I put an ad when I start working on a place, I get a lot of foot traffic and a lot of promises to come back in a couple of weeks to see the progress - almost never followed up.

Whether it is Google My Business (formerly Google Places), or driving traffic to your local site (or both preferably), you are putting yourself where the sellers are looking. 

I just installed Allure Traffic Master (easiest floor to install ever) from Home Depot.