Yes. Walking can support brain function, but the effect depends on what exactly constitutes an "improvement." A single walk can temporarily improve certain measures of attention, executive functions—the ability to plan, switch between tasks, and suppress unnecessary reactions—and especially the generation of new ideas.
Exercise does affect the brain: regular activity is associated with more favorable cognitive aging, can improve memory and executive function, reduce symptoms of anxiety and depression, and impact vascular and molecular mechanisms involved in neuroplasticity.
After an illness, don't immediately return to your previous training volume and intensity. For a mild respiratory infection, the modern approach isn't to wait a fixed number of days, but to start with a small test load and assess the body's response during it, immediately afterward, and in the following 24 hours.
Shoes do change the way you walk: the position of your foot when it contacts the ground, the length and frequency of your stride, the movement of your ankle and knee joints, the distribution of pressure under your foot, the work of your muscles, and, in some conditions, the movement of your pelvis and torso.
For most strength training, running, and team sports, it's best to use a general warm-up and dynamic movements before the main workout, saving longer static stretching for the end of the session or for a separate flexibility workout.
No, stretching every day is generally not necessary for health and maintaining normal flexibility. For most healthy adults, regular stretching 2-3 times a week is sufficient to maintain or gradually improve range of motion.
A foot massage mat can be useful as a simple means of providing pleasant sensory stimulation to the sole of the foot, a short-term massage sensation, and as a surface for some balance exercises.
Without shoes, the foot works more freely under support, the toes are less restricted by the toe of the shoe, the length and frequency of the step, the distribution of pressure and muscle work change.
