My first step would be to call the city or county engineering department and explain the situation to them. Frequently, new construction impacts storm water runoff and this is their purview. At the very least, they may assist with cleaning out the culvert (it doesn't look like there's an issue with standing water currently, but that may just be the angle of the photo). If they determine the problem rests entirely on your shoulders, then my next step would be to consult with a drainage engineer. Modifying drainage, especially when it effects wetlands, is not a good project for a homeowner without professional advice. The repercussions could be very expensive.

Hmm...we have our own water problem here we're trying to figure out, but yours seems a bit more solvable than mine! It seems that maybe to talk to the city could be a solution, especially since you said that when they put sidewalks in, it has made your problem worse. You may be able to have them regrade the area near the sidewalk so that the water flows onto the street and into drainage pipes properly. If not, then back to your plans. If the water is not backing up into your yard (and nearing your foundation), but rather just pooling in the dirt area shown in your picture, I would think about planting some shrubs that are water loving and would soak up puddles quickly. Maybe something like a dappled willow? I'm still learning a lot about plants, but someone else here who is much more knowledgeable could guide you better I'm sure! Or you could do what's been suggested to me--a dry river bed (but mine would be a 15 foot wide one with really substantial rocks, so not a few hour DIY project!)If you filled that are in with small river rock and then did some plantings, it would be pretty and multipurpose as well. Just a few thoughts. I would call the city first though to see what they say.

If the concrete that slopes toward the house is not removed and the area regraded, there may be some things that can be done to minimize the bad effect (such as Tozmo's patio above), but they will never be a complete or sure cure. I was once faced with replacing the main sewer drain of a house I owned and it necessitated the removal of the driveway -- only the good half of it, as the rest was junk. I broke it up in large chunks and used it in the back yard to make a sizable retaining wall. If you happen to have the need for a retaining wall, consider if removed concrete can work for you. My only regret about it was not scoring it with a saw into 18"x18" chunks. Some were massive and extraordinarily heavy. Would have been easier to use if there was some uniformity to the sizing. Whenever faced with major work or expense, evaluate the length of time you expect to be in the house. If less than five years, I'd work hard to find the cheaper, less expensive solutions. Where center lines of the sunroom door and brick path cross looks to be about 15' away from the door. For that distance, point B needs to be at least 2" lower than point A. Point C looks to be about 20 or 25' away. It needs to be 3" lower than point B. The drainage path center line would carry on dropping like that throughout the length of its extent. The brick walk coming from the back looks like, as it is, it's almost acting as a modified flume already. If its path continued (not necessarily out of brick) and it swung to the right of the garage, it could transition into a flume along the garage and be the main drainage path. The walk between the building I presume slopes toward the front ...? It should be dished out (where it's sloping away from each building toward the path's center line) to act as a flume. Water would be draining at each side of the garage.) The dark blue line represents the center line of the drainage path. The light blue lines represent the direction that water is travelling in order to feed the main path. i.e., they represent the slope. All of the slope that makes up drainage wold be blended and gradual, not sudden or abrupt, even though it might look like that in the concept.

Have somebody there who knows about drainage come and look at the situation in person.