If the water marks are higher than the ground outside, that means the water is coming from higher up. If the gutters are clogged (even two floors up), that could certainly be the source Heavy rain will find it's way behind the fascia, especially without a drip edge under the edge of the roof) and can end up in the basement. Check with a flashlight if your basement ceiling joists are wet in that area to confirm this.

Water can also come from a window well that doesn't drain properly. During the next deluge, see if water comes up to the window. If so, it surely is coming through the window frame.

From the looks of it, wall to wall carpet is a bad idea in your situation. Some options are remnants (paint..after proper prep... whatever doesn't get covered by the rug.) or tile. The last thing you want it deal with wall to wall that gets soaked along with the pad - that's a nightmare. It's hard enough dealing with a wet remnant, but much more tolerable.

As an update, I managed to get 5 dricore tiles up and mopped up the water that was under there. It really wasn't much in the end. Left it overnight and it was bone dry by the morning.
By chance, we had an appt with a landscaper today and he brought his plumber along to evaluate ways to divert the sump outflow. They advised me that that cement was not wet but that the wall next to it had been painted.
Also, on careful inspection, it appears that the only part of the basement to get water is immediately underneath the front cement stairs. We found a crack where it meets the house and it is slightly pitched toward the house. Seems that water made its way through the crack and down.
Plan now is to rip out the stairs and make sure that the foundation at that point is intact/sealed. Then rebuild them. We may put in a french drain along that wall as the whole front of the house will be getting ripped up to remove shrubs and other plants.
E