If you are laying tile then use the cement based boards.I am suggesting you safely encapsulate the existing vinyl tile and leave it alone.

There one flight of stairs to get into the unit.

Can anyone recommend an investor-friendly handyman that does small cement repairs around Baltimore?

Painted the down stairs, and added new flooring.

What the contractor promised was that he would use jack posts for the support beam and put down two cement columns and then insert a permanent thick wooden or steel beam once the wall was removed and columns set.

Is there any problem with the stairs increasing damages from moving furniture up and down them?

The landlord was not aware of this issue, you who live there and park there daily was unaware, obviously it wasn’t an issue until you parked over top of the cement parking barrier.  

And if you're trying for a variance, I would have a backup plan in place if you don't get it, I do agree that it's tough to get from what I've heard.Actually I have done a basement unit but it was in a quadplex and the building had two stairs, one in front and one in back, so two means of egress.

One thing is for sure and that is there is no elevator to success, everyone has to take the stairs 

Should I keep or place with traditional stairs?