Lower back pain when worse when standing should I continue going to work, I work on my feet all day and the pain just gets worse as day progresses?

Back pain in young. Dangerous causes of pain are unlikely in people age 18-50 years. So no immediate need for investigation unless other medical conditions like trauma, infection, immune compromise exist. Pain in the back is usually related to muscle/tendon issues or posture. Stress fracture is a common problem (5%) and can cause pain. Sacroiliac less common. Consider pelvis or abdominal sources. Md eval if worsens.
Yes, but get exam. Back pain can be of many sources. Obviously, your back itself, but also related causes are often biomechanical starting from the foot to the back and hips. I would first see an orthopedist or chiropracter for an exam and x-ray and go from there. Very often a good pair of custom foot orthotics can reduce or eliminate back pain if its origin is biomechanical and related to your gait.
See your doctor. Sounds like spinal stenosis if it is chronic (longer than 3 months duration). If less chronic could be a strain pattern. Which may be from an acute injury. This warrants seeing your physician to help determine the cause and start the diagnostic work up. There are some more serious diagnoses that include kinds of back pain.