I had a sidewalk up against the side of the house that was sloped toward the house. We got water in the basement a good 25 feet away from this sidewalk for many years without putting 2 and 2 together (but that's another story!). I had estimates of thousands of $ to put a drainage system in the basement. Instead, I got smart and removed the sidewalk and put a pipe in the ground to take water from the gutter out to the street. Bone dry for 5 years even during hurricane weather.

However, my house is higher then the street. Is it possible to remove the sidewalk and regrade to allow water to pass around to the back of the house?

Just some ideas.

The ground should slope down from the house such that at 10 feet away, it has a fall of 6 inches. Here's how they use swales to control water hilly neighborhoods.

The neighbor's runoff is not supposed to flow onto the neighbor's lot, so they built a swale on his property leading water to the front curb. Then the downhill neighbor's lot has a hill which needs to drain so there's another swale. On the back side of the block wall is a drain that leads to the hole in the curb.

If you were to do that you'd find the soil level at the house, measure out 10 feet, dig down to a point 6 inches below the house soil level, and contour the soil on both sides of that low point.

Side note: I need to do that at my new house. We had ankle deep water in the garage last Saturday night.

A french drain may help some. But you may get better long term results from regrading the land, as stated by daniel & dc. Looking at the picture your sidewalk is a bowl. The brick (?) border at the rear (near the house) and the raised soil/grass in front traps the water on the walkway. The water has nowhere to go until it turns that corner and soak into the mulch. It's hard to see but it looks like that mulch area near the corner is depressed. It may be from erosion & saturation. It also looks like all the water from the gutters flows backwards towards the sidewalk exacerbating the problem. Those drain pipes should be rerouted asap if that is the case.

You may be able to partially correct the problem by adding a perforated pipe a long the side walk in front, and then connect it to pipe that runs down the side of the yard between the side walk and house. You will need to make sure that it has some slope in the pipe. Use the black corrugated type.

You definitely need to get those down spout piped over to the side of the house where there appears to be a slope toward the back. You are collecting all of the water from your roof and putting it back on your foundation. I would run a pipe down the front of the house that collects the water from both downspouts. It could pass under the sidewalk or run down the side.

You may also ask a couple landscape contractors to make recommendation for solutions. Ask for a couple of estimates for solutions and cost.

Extend 2 downspouts to a lower elevation at least 10' from foundation. The ends of the pipes should be 100% open at the end with no grass build up. The pipes should be schedule 40 PVC with clean out ports attached at downspout/pipe connection.

Next,grade away from the foundation at 4% by using a clay soil. compacted very well. It looks like you will need a window well for the vent on the foundation

You may need to remove/replace the concrete sidewalk for the grading to take place.

An experienced landscape company would be able to install the components of the drainage system for you. Rely on swales and grading (not yard drains or perforated pipes) if possible.