This isn't a downspout problem, this is a grading problem. The ground has either eroded over time or was never properly graded in the first place. Probably a combination. Either way, the end result is that water collects in the low places and only dissipates by absorbing into the ground (not good; that can cause foundation problems like liquefaction of the soil underneath a slab or P&B foundation, or weeping and frost heaves against a basement wall) or evaporating.
You're not going to fix this by extending the downspout; you may reduce the amount of water collecting, but rain will still fall naturally into this valley next to your house, or fall against the side of your house and then run down into it. You're going to have to solve the problem by creating a continuously-downhill path for this water, directing it away from your house.
This sidewalk; is it sloped in a V with the lowest point at its center? If so, it was probably built as much to direct runoff as to be a walking path, and you should investigate where the sidewalk should be sending the water (probably around to the front of the house and down the driveway) and make adjustments accordingly. If not, why does it slope? It is really old and sloping due to erosion underneath it, or did the contractor who put it in just not have his level with him that day?
Can we get some real pictures of the side of your house and this valley, with indications of where the water currently pools?
Proper drainage depends on a constant gradient. These flexible pipes don't provide that.
Those concentric corrugations and the flexibility that allows them to follow ground contour with a portion of the length dropping into a depression or having to go over an obstruction leaves a pool of water that freezes.
Elbows and rigid downspout stock to get the roof runoff out away from the foundation works best. There are fold-up metal/vinyl 48/60" extenders available that allow for flowerbed maintenance.
If you have a lot of winter precipitation as rain, or a lot of melt-water, you need to get the runoff out and away from the house foundation, which is the purpose of a proper downspout system. Thus, extending the outflow of the downspouts is needed year around to prevent this.
Best Answer
In many places it's not legal to connect a downspout (storm water drainage from a roof) to a sanitary sewer system. The reason is easy to understand: if storm water flows into a sanitary sewer system the sewage treatment plant may easily be overwhelmed by rainfall. What happens when the sewage plant is overwhelmed? Untreated sewage escapes somewhere. Either the plant's output is not treated as it should have been or sewer plumbing upstream overflows.
On the other hand, if you're connecting a downspout to some kind of local storm water holding or drainage system, I suggest there should be a gap between the two near (above) ground level. If the underground drainage system should back up for any reason it's better for the storm water to spill out at ground level than for it to back up into the downspout and gutter system. By "gap" I mean use loose-fitting adapters so that backed up water can easily drain/spill out.
Gutters aren't always secured well; the weight of being filled to the top with water may tear them from the building. The backup may also encourage water to be wicked up into the roofing system. If the weather is cold enough for the standing water to freeze in the downspout and gutter, that brings a whole other raft of problems.