Insulation, Radiant and Vapor Barriers - Crawl space drawing cold air from fireplace?

I have a split level home - entry into main level w/ living room and fireplace to left...stairs to upper level w/ kitchen/hall/bedrooms…stairs to lower level w/ family room/den/laundry walk-out, half below grade. My crawl space is under the living room - access is via the den on lower level. It’s your typical vented crawl space, with two un-insulated exterior walls…and insulation on the other interior walls and ceiling (living room floor). However, I did find that the set of stairs going to lower level are fully exposed. I stapled and taped foil insulation to seal this off. Q1, is this normal for stairs to not be insulated from crawl space? Q2, is my insulation job the right way to address? And this issue might be contributing to my overall problem… I've found that at the very bottom of these stairs to lower level, where hard-wood stair meets ceramic floor, cold air is being drawn under the trim - and pretty sure it's coming from the mostly unused fireplace. The fireplace is on far left end of house. Air draws from there, goes in front of entry door and down the stairs. I’ve always thought the problem was the front door, but using lighter test, finally figured out that air was being drawn into crawlspace via bottom of stairs. It’s a wood-burning fireplace with heat circulating system (draws at bottom, goes through cavity, blows out at top). It also has outside air venting - which I think might be my problem since the knob to close this doesn’t seem to be doing anything - plan on investigating today. Q3, should I caulk or seal the bottom of my stairs? Or is the overall problem the fact that I have a fireplace letting air in from outside? A couple of other notes. I do have some duct work in crawl space, but it seems to be sealed well and insulated. I have heat pump. Air handler is on lower level…I’ve added a cold air return vent that I open in winter. I’ve read everything out there about encapsulating crawl space, which a long term solution, but I’m looking for sort term issue to my drafty main level/stairs. Thanks for any insight. I'm ready to troubleshoot until this issue is put to bed.

I was just reading about how fireplaces are not exactly energy efficient.
If I am not mistaken it was in the Money Stretcher blog.
The same vents provided to let out the smoke, will let in cold air.
Based on what you describe seems to be the source of all your problems.
Crawl space encapsulation might help improve the overall energy efficiency.
However the fireplace, when used, needs to draw air to fuel the fire and let out smoke.
I would be very cautious sealing a crawl space if that is where your fireplace draws its air supply from. You don't want to turn your house into a smoker, a carbon monoxide death trap or a fire hazard.
So here are two ways I think you can fix the problem:

Get a fireplace expert to look at the problem. Maybe install a new, more energy efficient fireplace or different vents. (I know nothing about fireplaces)
Get a crawl space encapsulation pro to take a look at the problem and see what can be done.

I will not try to tackle this on my own, based on advice from board members who are not able to inspect your place and evaluate the problem.