How to keep the carpet edges from curling upwards

Lay a very damp, but not dripping, heavy towel over the mark, then iron it with a regular clothes iron on high heat. Go over it for a few minutes until it's good and steamy. Then remove the towel and brush the carpet with a stiff brush, against the direction in which the fibers are lying down. Try to get them raised as much as possible and then let it dry.

There is double-sided "rug gripper" tape available at Target and Home Depot (probably most other stores, but I know those two carry it).

We use it to keep our entry rugs stuck to the tiled floor. It holds well enough that we can vacuum the rugs.

Disclaimer: Not sure how well it will stick to a carpet vs a hard floor.

Alternatively you could use rug pins/rug anchors to hold the area rugs to your carpet.