Not sure why you would ever have to keep an opening there. If your sump pump fails and the pit overflows the carpet is much easier to replace than the drywall and all the disintegrating thin trim. It doesn't look like you can restretch the low pile carpet so maybe a thin wood trim piece painted the same as the floor color might make it more invisible.

What about the rotted wood trim? That would bother me too. I suggest a piece of PVC trim board to cover up the ugly under the step. PVC trim comes in numerous shapes and sizes and can be painted and won't rot, and will cost you about 5 bucks.

for the ugly on the door casing I would suggest cutting the casing off and installing Plinth blocks and do the same for every door casing. (picture below) Or a plastic wood filler then sand smooth and paint.

I bought my present home in Feb 97 and the outside sump pump worked but the septic system was a leach tank full of stone with holes in the sides . We had a lot of snow during the winter, and that spring we had unusually high temperatures causing the snow to melt quick., this was the spring Grand forks N.D. flooded and burned. The ground saturated and poopy water backed up through the floor drain and basement shower drain. with no chance the sump could even keep up with the rising ground water level.

I was lucky because when I bought insurance they offered sewer backup insurance for an extra $15 so I bought it. Insurance paid off and I proceeded to rip out every stick of wood in the basement, even dried out it still was contaminated with poop water. My stairs are carpeted but the first riser has the carpet ending about 3/4 of the way down and trimmed out with PVC to match the baseboard which is also PVC, and this is what I'd do in your case.

No wood touches the floor, not even treated. I first glued and screwed down 1"x4" PVC boards, then put down treated 2.x4's bottom plate boards on top the PVC anchoring them with concrete anchor screws. Sheet rock ends 1" above the finished floor and all baseboard is 3" PVC. All door casings have a Plinth block at the bottom like in the picture below which is also made from PVC.

I also installed a secondary sump in the well room as backup if the primary fails, and it did once. No way was I going through this again, and the water needs to rise 3/4" before touching treated wood and 1 3/4' to touch sheet rock. I also replaced the septic, dug down to the footings to water proof the exterior walls and install a drain tile system. I also replace just about everything else on the house, so I bought a money pit to basically get the land, with a very high water table and can hit water 4' down. There are no full basements in my area, you either build up the ground for a walkout or build a split level 4' into the ground.

From someone who just had heavy rains and failed sump pumps last weekend, cover with a trim piece and get a WiFi water detector. They are only $20 We also had a back up pump and we didn't know both went bad.