This was quite an interesting read and you sent great documentation - I need to do the YouTube thing myself! I'm sure there will be several folks reading and responding with answers for all of the different parts of this question.
Most important to start with is the drainage. You can dig the French drain if you know where everything is. I checked, the number to call to have your utilities located is
811. This statewide service is called
Lonestar 811, and they send out various utility representatives to mark (they use different colored spray paint and flags for each company) for
free. Give them a few days to get out there, and then preserve the marks while you're working.
For mosquitoes, get a package of mosquito dunks over at Home Depot (least expensive place I've found to get them) and toss one in each puddle. If they dry out, no problem, but they get wet again they'll work again. And get a large container of granulated garlic (Sam's Club has it for about $20 and Costco is probably the same). The video no longer plays, but you can see the first photo in an interview on Channel 4 when Howard spoke about sprinkling garlic around the house and yard to repel mosquitoes. Take off the red lid but not the safely inside lid. Take a pencil and stab it into that safety liner 15-20 times and make a huge garlic shaker. You don't need a lot to do a good job, just shake garlic around near the doors and windows and around the yard. I have a jug of this and it lasts for several applications. I usually buy a new one every couple of years.
As you do any digging and weeding establish a spot for a compost pile, away from the poor drainage because you don't want standing water to leech out your nutrients. I have a large yard so I have three piles going at a time, the one I'm using, the one I'm contributing to, and the one from last year that is still breaking down. If you don't have as much room turn it over every so often and make sure to water it and you'll have great compost in a few months. Compost is going to be your most important tool out there.
Your best bet for a good overview of techniques is to visit the
organic guides and watch those videos. Bed preparation talks about those raised beds you want, but amendments broadcast to the yard will help with the soil. Lavasand, green sand, compost, and probably zeolite to help lock up some of the chemicals that have been used out there in the past. Look up each of these words and any others you come across in the
Natural Organic Library.
To sum up the organic part of this answer, I think the start I would use is to broadcast a mix of amendments and a robust application of dry molasses. And look into
zeolite. If you can tolerate the geckos, they'll do good work for you on bug control. Using the library link above you can also find information about bug control in the house.
Welcome to the group!