Ask Mr. Smarty Plants

Mr. Smarty Plants is sorry to tell you that getting rid of all these invasive weeds is not going to be easy.  Just a few days ago I wrote the answer to a question about getting rid of the non-native Cynodon dactylon (bermudagrass) in a lawn.  Bermudagrass is especially difficult because it spreads by seeds, stolons (above ground stems) and by underground rhizomes.  The advice in the previous answer stands for your bermudagrass. Sorghum halepense (johnsongrass) is another non-native invasive that is able to spread by two methods similar to bermudagrass—by seeds and by underground rhizomes. 

By stickers I assume you mean sandburs.  There are several species of sandburs, but I suspect you have Cenchrus spinifex (coastal sandbur).  It is important to remove the seeds (these are within the stickerburs) before they drop to the ground.  Since this is an annual species, it must drop its seeds to come up again.  Here is more information about Cenchrus sp. from California and ways to control it. You can also dig out and pull up the plants, but the most important thing is to not let the burs fall to the ground.  Digitaria sanguinalis (tall crabgrass) or Digitaria ischaemum (small or smooth crabgrass) also reproduces by seeds so it is important to remove seed heads before they mature.  This is also important for all your invasives—don't let them go to seed!

For all the invasive species you named you will need to use multiple control measures consisting of physically removing as many plants as possible (is there a teenager in your neighborhood who needs a summer job?), making sure that all the seed heads are removed before they mature, and judiciously using herbicides (see the suggested applications in the different control articles linked above for guidance).  You will need to be persistent and diligent.  If you think that this is too daunting, you can always use solarization to kill everything and start completely over.  The Native American Seed website has a very good description of how to do this in their article, Planting Tips for Native Grasses.  Here is another take on Soil Solarization from University of California-Davis.

There is some evidence that amphibians (e.g., toads, frogs and salamanders) are negatively affected by one of the most widely used herbicides, glyphosate, and the particular surfacant that is used to carry it. The risk could be lowered by painting on the herbicide with a small brush or sponge applicator rather than by spraying it.

For mowing, you will want to wait until most of your wildflowers have set and dropped their seeds.  You might find the How to Article, Meadow Gardening, helpful for advice about maintaining your meadow-like lawn.  Also, you don't want to mow too short—leaving your grasses and wildflowers several inches high (at least 4 to 6 inches) can help control bermudagrass and crabgrass since neither plant grows as well when shaded.

Finally, the Greater Madison Healthy Lawn Team (Madison, Wisconsin) has a very good treatment of Weed Control Methods using corn gluten, homemade herbicides made from liquid soap and vinegar, and solarization.

Good luck!