Do I need to bind the edges of a remnant if it has carpet pad attached on the underside

You absolutely should (must) bind the edges.

Many consumer rugs are cut to length from a roll stock and bound for sale. Binding the edge is how it transforms from a textile on a roll into a rug.

You could do it yourself if you had the requisite equipment and sewing skills but I can't imagine it would be tremendously expensive to have done.

You can find a video of a product that can be used to bind your own edges, as well as basic instructions on how to use it right here.

A summary of the video is below:

Tools you'll need

• Hot glue gun
• Scissors
• Straight edge

Procedure

Start by trimming the rug to provide a clean smooth edge. You can do this using a straight edge and a pair of scissors.

Next you'll want to trim the binding, so you start with a clean straight end.

Pull back the backing paper, and apply the binding starting at the middle of one edge.

When you get to the corner, use the edge of the rug as a guide to make a cut through the flat part of the binding. Then bend the binding around the corner, and continue around the rug.

Once you've reached the place you started, use the beginning end as a guide to cut the binding to the proper length. Trim any excess strings, then attach the ends using hot glue.

Run a 1/8" bead of hot glue, between the edge of the rug and the piping. Do this in small sections, so you can hold the two together until the glue sets. Continue all the way around the rug.

Alternatively, any rug shop or commercial carpet establishment should be able to do it for you...

The problems you describe are likely on the outside of the building. I'm willing to bet the landlord is unaware of this, or they are simply unwilling to fix it. This likely isn't fixable by you.

The moisture is coming through the concrete. Concrete is porous, so it will wick water from the surrounding soil naturally. To help prevent this process, basements are typically surrounded by a weeping tile. This is how the outside of the building basement should work

In new construction, this is corrugated pipe with a sleeve to minimize dirt, but in older buildings this was literally clay tile. So, what happens over time is this tile can break (it's ceramic), blocking the weepers. There's no way to find this out, except to excavate the outside wall and look at it. (image source)

My bet is

1. The weepers are broken
2. The wall has no waterproofing
3. The ground does not slope away from the building (you can confirm this visually)

Repairing this is not cheap, obviously. It's not surprising that it was ignored if it was known about.

If the joists are spongy, it means they've absorbed a lot of water. I'm surprised it hasn't attracted termites yet. This could be making the apartment structurally unsound if it has been an ongoing problem. Even if there are no termites, the wood may begin to rot.

The only thing you can try yourself is a dehumidifier. But if the room is humid enough to wet the floors and joists, a dehumidifier will likely be overwhelmed very quickly (you will have to empty the collected water frequently). The owner should call in a foundation waterproofing company to excavate the basement walls, water proof them, install new weeping tiles, and slope the ground away from the structure.