What is this weed?

Hello, I recently purchased my first home this past fall. Now that we made it to through winter to the spring, we have come to notice that our backyard does not drain well at all. Our yard is not level and we get ponds that last a week in the low spots. We have been looking into different options but being I just graduate College I'm on a budget. I just had a couple landscaping companies come out to quote how to fix it. One said to grad the lawn, dig a trench and lay a drain out to the street. It would cost $5000. Another came out and said they would "Bore the lawn" 7-14 times throughout the yard and fill with Pea gravel, then lay 17 yards of top soil, then seed. This would cost $2100. I like the second idea but wanted to know if this would work. It also seems pretty easy for a DIY. Just need steps on what I need to do. My other thought was to slope the dirt into a rain garden, if that would work. If you have any other ideas or can help me with the steps needed to do this myself it would be much appreciated. The Yard size is about 1000 sqft. And my property seems to be higher than our neighbors.

I recently moved into a home that has a 15-station sprinkler system. I've never had a sprinkler system before. When I first moved in, I tested the system and I found that zones 4 and 13 don't do anything. It may be just that they accidentally skipped over these in the distribution of the wiring. Even though there's wires hooked up at the controller for those stations, I don't think I have any heads that aren't working. But they're really hard to see unless they pop up. I have some detective work to do on the wiring. Fast forward to this week. I had natural gas brought up to the house. It's a 460-foot run that follows the driveway and then cuts under it to get to my house. Guess what runs along the driveway all 460-feet? Sprinkler heads. They cut through the piping for 6 sprinkler heads, the feed line to the box with the valves in it for that part of the property (5 stations) as well as the electrical cable that feeds said box. I repaired everything and am now left with 2 issues: 1) One zone does not come on. I have power to the solenoid - I can hear it click on and off. When I open the bleeder screw on the valve I get a weak spray from the 3 heads connected to that valve. If I twist the solenoid (It says "1/4 Turn On" I don't get anything from the sprinklers. If I do the same on any other zone, I get a normal spray. So what's going on here? Debris in the valve? 2) The last, lowest valve in different zone drains when the system is shut off. It's a the bottom of a very steep driveway so I figured it was normal that it couldn't fight gravity. However, after doing some reading it seems that this shouldn't be happening. I read that debris in that valve could case this issue. And in thinking about it, couldn't an underground leak also cause this? Vacuum can't hold the water back because air is getting in somewhere? This particular sprinkler head is on a zone that was cut in 2 other places but I repaired them myself and they were fine before I buried them. This one is an issue because when it drains it's washing away my topsoil and seed. [B]EDIT: I answered my own question. I replaced that sprinkler head and didn't use one with a drain-back valve. That's the problem![/B] Ideas and/or advice would be greatly appreciated!