Washing soda is a heavy duty version of baking soda. It's what gets clothes and dishes super clean. It's eco-friendly but not non-toxic if ingested (like if you ate it, not from cleaning dishes).
You can quite easily use up what you have for the dishwasher or laundry machine and then switch to baking soda if you'd like. You may end up liking the washing soda. Just use gloves when you use it.
As for pouring vinegar and baking soda at the same time, the purpose of the baking soda (basic pH) is to clean the oils and dirt off clothes. White vinegar has a low acidic pH, which help loosen fibers of clothing, so it acts like fabric softener - and also rinses any residues off regular soap or extra baking soda.
If you put them together, vinegar and baking soda produces fizz as a result of carbon dioxide bubbles forming. This resulting buffer solution can be helpful for other types of cleaning, like mopping floors, but it may be counterproductive for clothes cleaning. pH is a big factor in getting clothes clean.
You can opt for a Downy ball to put your vinegar in, should you choose to want something that acts like fabric softener. Otherwise, no need for it.
I've also noticed that a full cup of baking soda helps me get my clothes cleaner if I'm not using detergent, than a half cup as other people have mentioned. We have hard water in Atlanta, so using a little more may be beneficial for places with hard water.
Ladies, when you are using baking soda only, do you just wash your clothes in cold water or do you boost the cleaning power by adding warm or hot water? I typically wash my clothes in cold water, unless it's something totally soiled or bacteria-ridden (soaking up dog accidents).