Annual Ryegrass is sometimes used and recommended as a cover/green manure crop but many of us think it is a "weed", an unwanted plant, because of its ability to self seed.
I would stick with Buckwheat, Cereal or Winter, or Field Rye, Wheat, Oats, the grains and stay away from the grasses.

With the ones that you mentioned, Cereal Rye, Wheat, Oats are they easy to kill with out herbicide? Would just mowing them kill them? I would be planting my corn in March or April, so I would need to kill the cover crop at least in February.

I have not experienced any regrowth in cover crops after mowing them, or tilling them in, although others have reported problems. The Rodale Institute developed a crimper to knock down cover crops and leave them in place as mulch and I have seen nothing to indicate those plants grew back after being flattened.
When using Rye, Wheat, Oat, or sometimes hay as mulch I have observed new growth from the seeds that do not always get reaped and this may be what some of those that have had problems have seen, not regrowth from the old plants but new growth from seeds.

Since this is my first time with cover crops during the fall/winter I will try small patches of Rye and Fava.

Do both of these have to be flowering in order to kill them by mowing? I will plant them in August, so I have no idea when they will flower in Zone 9.

I live in South Texas and would like to start planting a cover crop. How soon and which cover crop is best suited for the Gulf Coast of Texas?

The cover crops you would seed in are the same as the ones I might seed in here in Michigan. There are the grasses (grains) such as Oats, Barley, wheat, and Cereal Rye. Some still talk about annual ryegrass which can become quite weedy.

There are legumes, the many peas and beans as well as clover that could add some Nitrogen to the soil as well as organic material.

Mostly what to seed in depends on what is readily, and fairly inexpensively, available in your area, although you can get whatever you want via the internet.

Perhaps this from Texas A & M may be of some interest.

http://articles.extension.org/pages/31141/cover-crops-for-arid-areas

kimmq is kimmsr

diamond, that is a tough question because you are in a climate which never freezes. much of the cover crop studies have been done for colder climates. Upthread I suggest favas, and they are good because they are nitrogen fixers, they have a tap root, and they die when you mow them. there are other crops that die by mowing, and you would be better off starting with those.

unless you can crimp your cover crops, in which case you can use difficult to terminate crops like rye triticale or vetch, which have a lot to offer in terms of the biology they can develop. navigate to youtube and search for stomper crimper. Mine has a steel edge but apparently this guy is doing well with a wooden edge. it is the small scale equivalent of a roller crimper, which you can also look at on youtube to get the basic concepts of terminating by crimping. time is of the essence for rolling crimping as crimping only works when plants are close to blooming.

glib I read an old thread where you said that you purchased or was purchasing 20 goumi seedlings. How did they turn out?

I purchased 20 seaberry seedlings. 13 are up and running. They would be 17 but my collaborator weedwhacked them, killing some (the rest restarted from the roots). They are also one year delayed.

Nick, this video series may be of interest to you, and may help explode some of the misinformation you might hear.

https://www.youtube.com/playlist?list=PL4J8PxoprpGa3wFYSXFu-BW_mMatleIt0

kimmq is kimmsr