I'll address your questions one at a time.

First, detergents are close to 100% effective at killing germs, but they can only be applied to exposed surfaces - any bacteria that have infected deeper tissue, or are hidden in nooks and crannies, cannot be reached by detergenty water, so other treatment is needed. Detergents cannot be applied internally - see the next part of my response for why. With that said, regular hand-washing is one of the most effective disease-control measures we have.

Second, detergent absolutely does have this effect on human tissue. Your skin cells in particular have evolved to be quite resilient, but detergent can and does kill them - fortunately your skin is made of many layers of cells, so damage is not immediately evident. Your skin also secretes oils to keep it soft and protected, and these oils use up a good amount of detergent before it can get to the cells themselves.
But wash your hands with strong dish soap many times a day, and after a few days you'll develop irritated, dry, cracking skin, with bleeding sores. This is also why the whole tide-pod thing was so dangerous - your internal tissue has not developed the same resilience as your skin has, and eating high-strength detergents can cause serious damage to your digestive tract.

Third, how can bacteria live on soap? Partly because of the reasons I mentioned in the previous paragraph, most body soaps have been developed to not so much kill bacteria and other cells, as dislodge them and allow them to be washed away. This causes minimal damage to your skin, but still cleans it - the bacteria have been washed off, rather than killed. Furthermore, those soaps and detergents that ARE lethal to cells (like dish soap) are only effective in the presence of water. See my original comment for more info there.

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