What the stores mostly sell for mulch is shredded bark and that is fine. Pretty much anything else organic you can get works too... grass clippings, fall leaves, shredded corn cobs, shredded paper, etc. You just want something that is going to cover the ground to hold moisture in and keep weeds down, and eventually break down to feed the soil. So yes a good layer of mulch does help suppress the weeds, though probably not 100%. And to keep suppressing weeds, it may need to be renewed mid-season.
Re the flower bed, my first suggestion would be to make it wider, to be in better scale with the house. Make the outer edge a bit curved and don't line every thing up like soldiers.
A mixture of a few shrubs and some perennials is nice. Shrubs might be viburnum, azalea, mahonia, serviceberry, coralberry, creeping juniper, or look around and what your neighbors are successfully growing and see what you like. Rosemary and lavender will grow into small shrubs for you and be lovely and fragrant. Perennials are too many to name and again you should look around at a good local nursery (not big box store) as well as talking to your neighbors about their gardens. But heres a few suggestions of southern plants that stay green most of the time (through milder winters) and have a long blooming season:
Chrysactinia mexicana (damianita)
Melampodium leucanthum (plains blackfoot)
Wedelia texana (hairy wedelia)
Glandularia bipinnatifida var. bipinnatifida (Prairie verbena)
Calyptocarpus vialis (straggler daisy)
Phyla nodiflora (frogfruit)
Plant a few perennials with room around them for them to spread and then fill in with annuals. Impatiens will get lush and beautiful in your climate and pump out flowers through a long season.